Simplicial spaces, lax algebras and the 222-Segal condition

Mark D Penney111 email: mpenney@mpim-bonn.mpg.de
Abstract

Dyckerhoff–Kapranov [5] and Gálvez-Carrillo–Kock–Tonks [7] independently introduced the notion of a 222-Segal space, that is, a simplicial space satisfying 222-dimensional analogues of the Segal conditions, as a unifying framework for understanding the numerous Hall algebra-like constructions appearing in algebraic geometry, representation theory and combinatorics. In particular, they showed that every 222-Segal object defines an algebra object in the \infty-category of spans.

In this paper we show that this algebra structure is inherited from the initial simplicial object Δ[]Δdelimited-[]\Delta\left[{\bullet}\right]. Namely, we show that the standard 111-simplex Δ[1]Δdelimited-[]1\Delta\left[{1}\right] carries a lax algebra structure. As a formal consequence the space of 111-simplices of a simplicial space is also a lax algebra. We further show that the 222-Segal conditions are equivalent to the associativity of this lax algebra.

1 Introduction

Simplicial objects satisfying the Segal conditions [26] are used throughout homotopy theory and higher category theory to encode coherently associative algebras. Recently, Dyckerhoff–Kapranov [5] and Gálvez-Carrillo–Kock–Tonks [7] independently introduced a generalisation of Rezk’s Segal conditions, the 222-Segal conditions. They showed that simplicial objects satisfying the 222-Segal conditions encode algebra objects in \infty-categories of spans. In this paper we shall elucidate the exact role played by the 222-Segal condition in the construction of these algebra objects.

Specifically, we provide a novel construction of the algebra in the \infty-category of spans associated to a 222-Segal object. We do so by exploiting the fact that the initial simplicial object in an \infty-category having finite limits is

Δ[]:Δop(inΔ)op,:Δdelimited-[]superscriptdouble-struck-Δopsuperscriptsubscriptindouble-struck-Δop\Delta\left[{\bullet}\right]:\mathbb{\Delta}^{\rm op}\to(\mathcal{F}{\rm in}_{\mathbb{\Delta}})^{\rm op},

where inΔsubscriptindouble-struck-Δ\mathcal{F}{\rm in}_{\mathbb{\Delta}} is the nerve of the category of level-wise finite simplicial sets. The following diagram

[Uncaptioned image] (1.1)

endows the standard 111-simplex Δ[1]Δdelimited-[]1\Delta\left[{1}\right] with a product μ𝜇\mu as an object of the \infty-category of spans in (inΔ)opsuperscriptsubscriptindouble-struck-Δop(\mathcal{F}{\rm in}_{\mathbb{\Delta}})^{\rm op}. While the product μ𝜇\mu fails to be associative, the diagram

[Uncaptioned image]

defines a non-invertible 222-morphism in the (,2)2(\infty,2)-category of bispans [10] in (inΔ)opsuperscriptsubscriptindouble-struck-Δop(\mathcal{F}{\rm in}_{\mathbb{\Delta}})^{\rm op} from μ(μ×Id)𝜇𝜇Id\mu\circ(\mu\times{\rm Id}) to μ(Id×μ)𝜇Id𝜇\mu\circ({\rm Id}\times\mu) which witnesses its lax associativity. We show the following:

Theorem 1.

The standard 111-simplex Δ[1]Δdelimited-[]1\Delta\left[{1}\right] is a lax algebra in the (,2)2(\infty,2)-category of bispans in (inΔ)opsuperscriptsubscriptindouble-struck-Δop(\mathcal{F}{\rm in}_{\mathbb{\Delta}})^{\rm op} with product μ𝜇\mu.

From any simplicial object 𝒳𝒞Δsubscript𝒳subscript𝒞double-struck-Δ\mathcal{X}_{\bullet}\in\mathcal{C}_{\mathbb{\Delta}} in an \infty-category 𝒞𝒞\mathcal{C} having finite limits one has a finite limit preserving functor 𝒳:(inΔ)op𝒞:𝒳superscriptsubscriptindouble-struck-Δop𝒞\mathcal{X}:(\mathcal{F}{\rm in}_{\mathbb{\Delta}})^{\rm op}\to\mathcal{C} given by right Kan extension,

(inΔ)opsuperscriptsubscriptindouble-struck-Δop\textstyle{(\mathcal{F}{\rm in}_{\mathbb{\Delta}})^{\rm op}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}𝒳𝒳\scriptstyle{\mathcal{X}}𝒞𝒞\textstyle{\mathcal{C}}Δopsuperscriptdouble-struck-Δop\textstyle{\mathbb{\Delta}^{\rm op}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}𝒳𝒳\scriptstyle{\mathcal{X}}

The simplicial object 𝒳subscript𝒳\mathcal{X}_{\bullet} is then the image of Δ[]Δdelimited-[]\Delta\left[{\bullet}\right] under the right Kan extension. As a formal consequence of Theorem 1 one obtains a lax algebra structure on 𝒳1subscript𝒳1\mathcal{X}_{1} as an object of the (,2)2(\infty,2)-category of bispans in 𝒞𝒞\mathcal{C}. We show that the 222-Segal condition is equivalent to the associativity of this lax algebra.

Theorem 2.

The object of 111-simplices 𝒳1subscript𝒳1\mathcal{X}_{1} of a simplicial object 𝒳𝒞Δ𝒳subscript𝒞double-struck-Δ\mathcal{X}\in\mathcal{C}_{\mathbb{\Delta}} is canonically a lax algebra in the (,2)2(\infty,2)-category of bispans in 𝒞𝒞\mathcal{C}. Furthermore, 𝒳subscript𝒳\mathcal{X}_{\bullet} satisfies the 222-Segal conditions if and only if 𝒳1subscript𝒳1\mathcal{X}_{1} is an algebra object.

To our knowledge there are two previous constructions of the above algebra associated to a 222-Segal object by different techniques and at differing levels of generality. The first is due to Dyckerhoff–Kapranov, who associate to each injectively fibrant 222-Segal object 𝒳𝒳\mathcal{X} in a combinatorial simplicial model category a monad on the (,2)2(\infty,2)-category of bispans which makes 𝒳1subscript𝒳1\mathcal{X}_{1} an algebra when 𝒳0similar-to-or-equalssubscript𝒳0\mathcal{X}_{0}\simeq\ast ([5] 11.1). Gálvez-Carrillo–Kock–Tonks showed the space of 111-simplices in a 222-Segal space carries an algebra structure using a novel equivalence between simplicial spaces and certain monoidal functors ([7] 7.4).

By first constructing the lax algebra for the initial simplicial object Δ[]Δdelimited-[]\Delta\left[{\bullet}\right] we are able to work at the highest level of generality possible, that of 222-Segal objects in an \infty-category having finite limits. Furthermore, this paper is the first in a series of papers which build new examples of higher categorical bialgebras using 222-Segal spaces [24, 23]. The construction given here provides the groundwork for these later papers.

Outline.

The paper begins in Section 2 with the necessary background for defining the notion of a lax algebra object in a symmetric monoidal (,2)2(\infty,2)-category. As the informal description of a lax algebra given above explicitly makes use of non-invertible 222-morphisms it should come as no surprise that the definition makes use of ideas which are inherently (,2)2(\infty,2)-categorical. Specifically, one needs lax functors between (,2)2(\infty,2)-categories. In the particular model of (,2)2(\infty,2)-categories that we use in this paper lax functors are defined in terms of the unstraightening construction, the \infty-categorical generalisation of the ordinary Grothendieck construction. A lax algebra object in a symmetric monoidal (,2)2(\infty,2)-category is then defined to be a symmetric monoidal lax functor out of the category 𝒜lg𝒜superscriptlgcoproduct\mathcal{A}{\rm lg}^{\amalg} which corepresents algebra objects. In Section 2.1 we review our chosen model of (,2)2(\infty,2)-categories, in Section 2.2 we define lax functors in terms of the unstraightening construction and in Section 2.3 we introduce the category 𝒜lg𝒜superscriptlgcoproduct\mathcal{A}{\rm lg}^{\amalg}.

As indicated in Theorems 1 and 2 we will be constructing lax algebra objects in symmetric monoidal (,2)2(\infty,2)-categories of bispans, which are the subjects of Section 3. One of the technical hurdles that one must surmount when working with lax functors of (,2)2(\infty,2)-categories is the difficulty of providing an explicit description of the unstraightening construction. As such, we devote a considerable part of this section to the unstraightening of the (,2)2(\infty,2)-category of bispans, rendering it into a form which is adequate for our purposes. We begin, in Section 3.1, with a brief discussion on the twisted arrow construction, a construction which appears throughout this work. Next we review Haugseng’s original definition of the (,2)2(\infty,2)-category of bispans in Section 3.2. Finally, Sections 3.3 and 3.4 contain technical material culminating in a description of the unstraightening of the (,2)2(\infty,2)-category of bispans.

While the previous two sections have been essentially preparatory, Section 4 presents the main results of the paper. We begin in Section 4.1 by proving Theorem 1, that is, by giving an explicit construction of a lax algebra structure on Δ[1]Δdelimited-[]1\Delta\left[{1}\right]. The first statement in Theorem 2 is proven in Section 4.2 as a formal consequence of the construction in the previous section. Finally, in Section 4.3 we prove the second statement of Theorem 2 connecting the 222-Segal condition to the associativity of the lax algebras constructed in Section 4.2.

Acknowledgements.

We would like to thank Tobias Dyckerhoff for a number of insightful conversations on the subject of 222-Segal spaces. We also thank Rune Haugseng for answering some of our questions regarding the unstraightening construction and the (,2)2(\infty,2)-category of bispans. Finally, we are indebted to Joachim Kock for his thorough feedback on an earlier draft of this paper.

Notational conventions and simplicial preliminaries.

Throughout this paper we make extensive use of the theory of \infty-categories as developed by Joyal [13, 14] and Lurie [19]. In particular, by an \infty-category we shall always mean a quasi-category, a simplicial set having fillers for all inner horns.

Ordinary, simplicially-enriched or model categories will always be denoted by either Greek letters (e.g. ΔΔ\Delta) or in ordinary font (e.g. C𝐶C). \infty-categories will either be blackboard Greek letters (e.g. Δdouble-struck-Δ\mathbb{\Delta}) or have the first character in calligraphic font (e.g. 𝒞𝒞\mathcal{C}).

For ordinary categories C𝐶C and D𝐷D, Fun(C,D)Fun𝐶𝐷{\rm Fun}(C,D) is the category of functors and natural transformations and Map(C,D)Map𝐶𝐷{\rm Map}(C,D) is the groupoid of functors and natural isomorphisms. Analogously, for \infty-categories 𝒞𝒞\mathcal{C} and 𝒟𝒟\mathcal{D}, un(𝒞,𝒟)un𝒞𝒟\mathcal{F}{\rm un}(\mathcal{C},\mathcal{D}) is the \infty-category of functors and ap(𝒞,𝒟)ap𝒞𝒟\mathcal{M}{\rm ap}(\mathcal{C},\mathcal{D}) is the largest Kan complex inside un(𝒞,𝒟)un𝒞𝒟\mathcal{F}{\rm un}(\mathcal{C},\mathcal{D}). In particular, the (\infty-)category of k𝑘k-fold simplicial objects in an ordinary category C𝐶C or \infty-category 𝒞𝒞\mathcal{C} are

CΔk=Fun((Δop)×k,C)and𝒞Δk=un((Δop)×k,𝒞),subscript𝐶superscriptΔ𝑘FunsuperscriptsuperscriptΔopabsent𝑘𝐶andsubscript𝒞superscriptdouble-struck-Δ𝑘unsuperscriptsuperscriptdouble-struck-Δopabsent𝑘𝒞C_{\Delta^{k}}={\rm Fun}\left(\left(\Delta^{\rm op}\right)^{\times k},C\right)\ {\rm and}\ \mathcal{C}_{\mathbb{\Delta}^{k}}=\mathcal{F}{\rm un}\left(\left(\mathbb{\Delta}^{\rm op}\right)^{\times k},\mathcal{C}\right),

where ΔΔ\Delta is the category of non-empty linearly ordered finite sets and the \infty-category Δdouble-struck-Δ\mathbb{\Delta} is the nerve of ΔΔ\Delta.

Let qCatqCat{\rm qCat} denote the simplicially-enriched category of quasi-categories with mapping spaces given by apap\mathcal{M}{\rm ap}. The \infty-category of \infty-categories, 𝒞at𝒞subscriptat\mathcal{C}{\rm at}_{\infty}, is the coherent nerve of qCatqCat{\rm qCat} ([19] 3.0.0.1). The full subcategory of qCatqCat{\rm qCat} of Kan complexes is denoted KanKan{\rm Kan}, and its coherent nerve, 𝒮𝒮\mathcal{S}, is the \infty-category of spaces. The inclusion 𝒮𝒮\textstyle{\mathcal{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}𝒞at𝒞subscriptat\textstyle{\mathcal{C}{\rm at}_{\infty}} admits a right adjoint ([19] 1.2.5.3, 5.2.4.5) denoted

():𝒞at𝒮.:superscriptsimilar-to-or-equals𝒞subscriptat𝒮(-)^{\simeq}:\mathcal{C}{\rm at}_{\infty}\to\mathcal{S}.

We denote by css the category of bisimplicial sets carrying the Rezk model structure [26]. This is a simplicial model category whose fibrant-cofibrant objects are the complete Segal spaces. The coherent nerve of the full subcategory of complete Segal spaces, denoted 𝒞SS𝒞SS\mathcal{C}{\rm SS}, is canonically equivalent to 𝒞at𝒞subscriptat\mathcal{C}{\rm at}_{\infty} ([15] 4.11).

The category of finite sets is denoted by FinFin{\rm Fin} and every object is isomorphic to one of the form

n¯={1,,n}Fin.¯𝑛1𝑛Fin\underline{n}=\{1,\cdots,n\}\in{\rm Fin}.

Similarly, FinsubscriptFin{\rm Fin}_{*} is the category of pointed finite sets. Objects of FinsubscriptFin{\rm Fin}_{*} are denoted Xsubscript𝑋X_{*}, with \ast the basepoint and X𝑋X the complement of the basepoint. The \infty-categories inin\mathcal{F}{\rm in} and insubscriptin\mathcal{F}{\rm in}_{*} are, respectively, the nerves of FinFin{\rm Fin} and FinsubscriptFin{\rm Fin}_{*}.

We adopt the topologists convention for the category ΔΔ\Delta of non-empty, linearly ordered finite sets in that we label its objects by

[n]={0<<n}Δ.delimited-[]𝑛0𝑛Δ[n]=\{0<\cdots<n\}\in\Delta.

The active and inert morphisms form a factorization system on ΔΔ\Delta. The former are those morphisms which preserve the bottom and top elements while the latter are the inclusions of subintervals. Active morphisms will be denoted by arrows of the form (|absent|\rightarrow\Mapsfromchar) while inert morphisms by arrows of the form (\rightarrowtail). Their corresponding wide subcategories are denoted, respectively, by ΔacsubscriptΔac\Delta_{\rm ac} and ΔinsubscriptΔin\Delta_{\rm in}. Every morphism in ΔΔ\Delta can be uniquely factored as an active followed by an inert morphism. The \infty-categories Δacsubscriptdouble-struck-Δac\mathbb{\Delta}_{\rm ac} and Δinsubscriptdouble-struck-Δin\mathbb{\Delta}_{\rm in} are, respectively, the nerves of ΔacsubscriptΔac\Delta_{\rm ac} and ΔinsubscriptΔin\Delta_{\rm in}.

Remark 1.1.

The active-inert factorisation of morphisms in ΔΔ\Delta is a particular example of the general notion of generic-free factorisations in the theory of monads as developed by Weber [28] and Berger-Mellies-Weber [4]. Following Lurie [18] and Haugseng [11], we adopt the former terminology as we feel that it is more descriptive.

The category ΔΔ\Delta is a full subcategory of Δ+subscriptΔ\Delta_{+}, the category of (possibly empty) linearly ordered finite sets, the objects of which are

n={1<<n}Δ+.delimited-⟨⟩𝑛1𝑛subscriptΔ\langle n\rangle=\{1<\cdots<n\}\in\Delta_{+}.

The category \nabla ([8] 8)222Note that our category \nabla is the opposite of the one defined in [8] has the same objects as Δ+subscriptΔ\Delta_{+} and morphisms given by spans of the form

kdelimited-⟨⟩𝑘\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\langle k\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}ndelimited-⟨⟩𝑛\textstyle{\langle n\rangle}m.delimited-⟨⟩𝑚\textstyle{\langle m\rangle\,.} (1.2)

The category of k𝑘k-fold nabla objects in a category C𝐶C is

Ck=Fun((op)×k,C).subscript𝐶superscript𝑘Funsuperscriptsuperscriptopabsent𝑘𝐶C_{\nabla^{k}}={\rm Fun}\left((\nabla^{\rm op})^{\times k},C\right).

There is a bijective on objects and full functor 𝔊:Δ:𝔊Δ\mathfrak{G}:\Delta\to\nabla which restricts to isomorphisms ΔacΔ+opsimilar-to-or-equalssubscriptΔacsuperscriptsubscriptΔop\Delta_{\rm ac}\simeq\Delta_{+}^{\rm op} and Δin1(Δ+)in1similar-to-or-equalssuperscriptsubscriptΔinabsent1superscriptsubscriptsubscriptΔinabsent1\Delta_{\rm in}^{\geq 1}\simeq(\Delta_{+})_{\mathrm{in}}^{\geq 1}([8] 8.2). Restriction along 𝔊𝔊\mathfrak{G} induces a fully faithful functor

𝔊:CkCΔk.:superscript𝔊subscript𝐶superscript𝑘subscript𝐶superscriptΔ𝑘\mathfrak{G}^{*}:C_{\nabla^{k}}\to C_{\Delta^{k}}. (1.3)

The category ΔacsubscriptΔac\Delta_{\rm ac} has a canonical monoidal structure

[n][m]=[n+m]delimited-[]𝑛delimited-[]𝑚delimited-[]𝑛𝑚[n]\vee[m]=[n+m]

having unit [0]delimited-[]0[0]. The functor 𝔊𝔊\mathfrak{G} restricts to a monoidal equivalence if we endow Δ+subscriptΔ\Delta_{+} with the monoidal structure

n+m=n+mdelimited-⟨⟩𝑛delimited-⟨⟩𝑚delimited-⟨⟩𝑛𝑚\langle n\rangle+\langle m\rangle=\langle n+m\rangle (1.4)

having unit 0delimited-⟨⟩0\langle 0\rangle.

Finally, throughout this paper algebra objects are assumed to be unital.

2 Lax algebras in symmetric monoidal (,2)2(\infty,2)-categories

In this section we cover the necessary background for defining the notion of a lax algebra object in a symmetric monoidal (,2)2(\infty,2)-category. We begin in Section 2.1 with a review our chosen model of (,2)2(\infty,2)-categories. In Section 2.2 we define lax functors in terms of the unstraightening construction. Finally, in Section 2.3 we introduce the category 𝒜lg𝒜superscriptlgcoproduct\mathcal{A}{\rm lg}^{\amalg} which corepresents algebras.

2.1 Review of symmetric monoidal (,2)2(\infty,2)-categories

The model of (,2)2(\infty,2)-categories that we use in this work is the one originally introduced by Lurie in [20].

Definition 2.1.

An (,2)2(\infty,2)-category is a simplicial \infty-category (𝒞at)Δsubscript𝒞subscriptatdouble-struck-Δ\mathcal{B}\in(\mathcal{C}{\rm at}_{\infty})_{\mathbb{\Delta}} such that

  1. 1.

    \mathcal{B} is a Segal object, that is, for each n2𝑛2n\geq 2 the functor n1×0×01subscript𝑛subscriptsubscript0subscriptsubscript0subscript1subscript1\mathcal{B}_{n}\to\mathcal{B}_{1}\times_{\mathcal{B}_{0}}\cdots\times_{\mathcal{B}_{0}}\mathcal{B}_{1} is an equivalence;

  2. 2.

    The \infty-category 0subscript0\mathcal{B}_{0} is a space, that is, 0𝒮𝒞atsubscript0𝒮𝒞subscriptat\mathcal{B}_{0}\in\mathcal{S}\subset\mathcal{C}{\rm at}_{\infty}; and,

  3. 3.

    The Segal space Δopsuperscriptdouble-struck-Δop\textstyle{\mathbb{\Delta}^{\rm op}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\scriptstyle{\mathcal{B}}𝒞at𝒞subscriptat\textstyle{\mathcal{C}{\rm at}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}()superscriptsimilar-to-or-equals\scriptstyle{(-)^{\simeq}}𝒮𝒮\textstyle{\mathcal{S}} is complete.

The \infty-category of (,2)2(\infty,2)-categories is a full subcategory 𝒞at(,2)𝒞subscriptat2\textstyle{\mathcal{C}{\rm at}_{(\infty,2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}(𝒞at)Δsubscript𝒞subscriptatdouble-struck-Δ\textstyle{(\mathcal{C}{\rm at}_{\infty})_{\mathbb{\Delta}}}. In other words, functors between (,2)2(\infty,2)-categories are simply natural transformations.

Remark 2.2.

A widely used model of (,2)2(\infty,2)-categories in the literature are 222-fold complete Segal spaces as introduced by Barwick [1]. Barwick’s model is recovered from the one used in this work by presenting 𝒞at𝒞subscriptat\mathcal{C}{\rm at}_{\infty} as 𝒞SS𝒞SS\mathcal{C}{\rm SS}.

Let op:ΔΔ:opΔΔ{\rm op}:\Delta\to\Delta denote the automorphism sending [n]delimited-[]𝑛[n] to [n]opsuperscriptdelimited-[]𝑛op[n]^{\rm op}.

Definition 2.3.

For an (,2)2(\infty,2)-category \mathcal{B} its opposite (,2)2(\infty,2)-category is the composite

op:ΔopopΔop𝒞at.:superscriptopsuperscriptdouble-struck-Δopopsuperscriptdouble-struck-Δop𝒞subscriptat\mathcal{B}^{\rm op}:\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 10.12224pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\crcr}}}\ignorespaces{\hbox{\kern-10.12224pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathbb{\Delta}^{\rm op}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 12.4278pt\raise 5.1875pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8264pt\hbox{$\scriptstyle{{\rm op}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 28.12224pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 28.12224pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathbb{\Delta}^{\rm op}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 51.88754pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.39166pt\hbox{$\scriptstyle{\mathcal{B}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 66.36671pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 66.36671pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathcal{C}{\rm at}_{\infty}}$}}}}}}}\ignorespaces}}}}\ignorespaces.

There is a universal way to extract from a Segal object \mathcal{B} in 𝒞at𝒞subscriptat\mathcal{C}{\rm at}_{\infty} a new Segal object 𝒰𝒰\mathcal{U}\mathcal{B} whose \infty-category of 00-simplices is a space: let 𝒮eg(𝒞at)𝒮eg𝒞subscriptat\mathcal{S}{\rm eg}\left({\mathcal{C}{\rm at}_{\infty}}\right) denote the full subcategory of (𝒞at)Δsubscript𝒞subscriptatdouble-struck-Δ(\mathcal{C}{\rm at}_{\infty})_{\mathbb{\Delta}} spanned by Segal objects and let 𝒮eg0(𝒞at)𝒮subscripteg0𝒞subscriptat\mathcal{S}{\rm eg}_{0}\left({\mathcal{C}{\rm at}_{\infty}}\right) be the full subcategory of the former spanned by those objects satisfying Condition 222 above. Then the inclusion 𝒮eg0(𝒞at)𝒮subscripteg0𝒞subscriptat\textstyle{\mathcal{S}{\rm eg}_{0}\left({\mathcal{C}{\rm at}_{\infty}}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}𝒮eg(𝒞at)𝒮eg𝒞subscriptat\textstyle{\mathcal{S}{\rm eg}\left({\mathcal{C}{\rm at}_{\infty}}\right)} admits a right adjoint ([10] 2.13)

𝒰:𝒮eg(𝒞at)𝒮eg0(𝒞at).:𝒰𝒮eg𝒞subscriptat𝒮subscripteg0𝒞subscriptat\mathcal{U}:\mathcal{S}{\rm eg}\left({\mathcal{C}{\rm at}_{\infty}}\right)\to\mathcal{S}{\rm eg}_{0}\left({\mathcal{C}{\rm at}_{\infty}}\right).

Explicitly, given 𝒮eg(𝒞at)𝒮eg𝒞subscriptat\mathcal{B}\in\mathcal{S}{\rm eg}\left({\mathcal{C}{\rm at}_{\infty}}\right), we have 𝒰0=0𝒰subscript0superscriptsubscript0similar-to-or-equals\mathcal{U}\mathcal{B}_{0}=\mathcal{B}_{0}^{\simeq} and for each n1𝑛1n\geq 1 a pullback square

𝒰n𝒰subscript𝑛\textstyle{\mathcal{U}\mathcal{B}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}nsubscript𝑛\textstyle{\mathcal{B}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}(0)n+1superscriptsuperscriptsubscript0similar-to-or-equals𝑛1\textstyle{\left(\mathcal{B}_{0}^{\simeq}\right)^{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}(0)n+1superscriptsubscript0𝑛1\textstyle{\left(\mathcal{B}_{0}\right)^{n+1}} (2.1)

There are a number of ways in the literature to define symmetric monoidal (,2)2(\infty,2)-categories. We follow Lurie ([18] 2.0.0.7) in choosing the one generalising Segal’s notion of a special ΓΓ\Gamma-space [27].

Definition 2.4.

A symmetric monoidal (,2)2(\infty,2)-category is a functor

:in×Δop𝒞at:superscripttensor-productsubscriptinsuperscriptdouble-struck-Δop𝒞subscriptat\mathcal{B}^{\otimes}:\mathcal{F}{\rm in}_{*}\times\mathbb{\Delta}^{\rm op}\to\mathcal{C}{\rm at}_{\infty}

such that

  1. 1.

    For each Sinsubscript𝑆subscriptinS_{*}\in\mathcal{F}{\rm in}_{*}, the simplicial \infty-category (S,)superscripttensor-productsubscript𝑆\mathcal{B}^{\otimes}(S_{*},\bullet) is an (,2)2(\infty,2)-category.

  2. 2.

    For each [n]Δopdelimited-[]𝑛superscriptdouble-struck-Δop[n]\in\mathbb{\Delta}^{\rm op} and Sinsubscript𝑆subscriptinS_{*}\in\mathcal{F}{\rm in}_{*}, the map (S,[n])sS({s},[n])superscripttensor-productsubscript𝑆delimited-[]𝑛subscriptproduct𝑠𝑆superscripttensor-productsubscript𝑠delimited-[]𝑛\mathcal{B}^{\otimes}(S_{*},[n])\to\prod_{s\in S}\mathcal{B}^{\otimes}(\{s\}_{*},[n]) is an equivalence.

The \infty-category of symmetric monoidal (,2)2(\infty,2)-categories is a full subcategory

𝒞at(,2)un(in×Δop,𝒞at).𝒞superscriptsubscriptat2tensor-productunsubscriptinsuperscriptdouble-struck-Δop𝒞subscriptat\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 18.21114pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-18.21114pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathcal{C}{\rm at}_{(\infty,2)}^{\otimes}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 18.21114pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.81117pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.81117pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathcal{F}{\rm un}\left(\mathcal{F}{\rm in}_{*}\times\mathbb{\Delta}^{\rm op},\mathcal{C}{\rm at}_{\infty}\right)}$}}}}}}}\ignorespaces}}}}\ignorespaces.

In other words, a symmetric monoidal functor between symmetric monoidal (,2)2(\infty,2)-categories is simply a natural transformation, that is, a morphism in un(in×Δop,𝒞at)unsubscriptinsuperscriptdouble-struck-Δop𝒞subscriptat\mathcal{F}{\rm un}\left(\mathcal{F}{\rm in}_{*}\times\mathbb{\Delta}^{\rm op},\mathcal{C}{\rm at}_{\infty}\right).

2.2 Symmetric monoidal lax functors via the unstraightening construction

Recall that a lax functor between ordinary 222-categories L:AB:𝐿𝐴𝐵L:A\rightsquigarrow B differs from a functor in that it no longer respects identity arrows and composition of 111-morphisms [16]. Instead, for each object a𝑎a of A𝐴A and each pair of composable morphisms f𝑓f and g𝑔g, one has (not necessarily invertible) 222-morphisms IdL(a)L(Ida)subscriptId𝐿𝑎𝐿subscriptId𝑎{\rm Id}_{L(a)}\Rightarrow L({\rm Id}_{a}) and L(g)L(f)L(gf)𝐿𝑔𝐿𝑓𝐿𝑔𝑓L(g)\circ L(f)\Rightarrow L(g\circ f) in B𝐵B witnessing the lax preservation of unitality and composition. Furthermore, these 222-morphisms must satisfy associativity and unitality coherence equations.

The notion of a lax functor between (,2)2(\infty,2)-categories requires the use of the theory of cocartesian fibrations of \infty-categories and the unstraightening construction as developed by Lurie ([19] 2.4). Recall that the unstraightening construction defines an equivalence between functors 𝒞𝒞at𝒞𝒞subscriptat\mathcal{C}\to\mathcal{C}{\rm at}_{\infty} and cocartesian fibrations over 𝒞𝒞\mathcal{C} ([19] 3.2.0.1),

Un:un(𝒞,𝒞at)𝒞ocart/𝒞.:Unun𝒞𝒞subscriptatsimilar-to𝒞subscriptocartabsent𝒞{\rm Un}:\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 31.56395pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-31.56395pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathcal{F}{\rm un}(\mathcal{C},\mathcal{C}{\rm at}_{\infty})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 30.64174pt\raise 4.28406pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.28406pt\hbox{$\scriptstyle{\sim}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 41.16399pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 41.16399pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\cal C}{\rm ocart}_{/\mathcal{C}}}$}}}}}}}\ignorespaces}}}}\ignorespaces.

Under this equivalence a natural transformation η:FG:𝜂𝐹𝐺\eta:F\Rightarrow G is sent to a morphism

Un(F)Un𝐹\textstyle{{\rm Un}(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Un(η)Un𝜂\scriptstyle{{\rm Un}(\eta)}Un(G)Un𝐺\textstyle{{\rm Un}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}𝒞𝒞\textstyle{\mathcal{C}}

such that Un(η)Un𝜂{\rm Un}(\eta) sends cocartesian morphisms in Un(F)Un𝐹{\rm Un}(F) to cocartesian morphisms in Un(G)Un𝐺{\rm Un}(G). The unstraightening construction is natural in 𝒞𝒞\mathcal{C} in the sense that from a composite

𝒟𝒟\textstyle{\mathcal{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}G𝐺\scriptstyle{G}𝒞𝒞\textstyle{\mathcal{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}F𝐹\scriptstyle{F}𝒞at𝒞subscriptat\textstyle{\mathcal{C}{\rm at}_{\infty}}

one has a pullback diagram ([9] A.31)

Un(FG)Un𝐹𝐺\textstyle{{\rm Un}(FG)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Un(F)Un𝐹\textstyle{{\rm Un}(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}𝒟𝒟\textstyle{\mathcal{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}G𝐺\scriptstyle{G}𝒞.𝒞\textstyle{\mathcal{C}\,.}

Taking 𝒞=Δop𝒞superscriptdouble-struck-Δop\mathcal{C}=\mathbb{\Delta}^{\rm op} one can unstraighten an (,2)2(\infty,2)-category, and functors become morphisms of cocartesian fibrations over Δopsuperscriptdouble-struck-Δop\mathbb{\Delta}^{\rm op} which preserve cocartesian morphisms. Lax functors will still be morphisms of fibrations but will preserve fewer cocartesian morphisms.

Definition 2.5.

A lax functor L:𝒜:𝐿𝒜L:\mathcal{A}\rightsquigarrow\mathcal{B} between (,2)2(\infty,2)-categories 𝒜,:Δop𝒞at:𝒜superscriptdouble-struck-Δop𝒞subscriptat\mathcal{A},\,\mathcal{B}:\mathbb{\Delta}^{\rm op}\to\mathcal{C}{\rm at}_{\infty} is a morphism

Un(𝒜)Un𝒜\textstyle{{\rm Un}\left(\mathcal{A}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}L𝐿\scriptstyle{L}Un()Un\textstyle{{\rm Un}\left(\mathcal{B}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Δopsuperscriptdouble-struck-Δop\textstyle{\mathbb{\Delta}^{\rm op}}

such that L𝐿L sends cocartesian lifts of morphisms in (Δin)opsuperscriptsubscriptdouble-struck-Δinop(\mathbb{\Delta}_{\rm in})^{\rm op}, the subcategory of inert morphisms, to cocartesian morphisms.

Remark 2.6.

We are not certain as to the exact history of this approach to defining lax functors between (,2)2(\infty,2)-categories. We first learned it from Dyckerhoff–Kapranov ([5] 9.2.8) and Lurie’s definition of a morphism of \infty-operads ([19] 2.1.2.7) is qualitatively similar. We believe an analogous statement must be known for lax functors between 222-categories but do not know any references.

Remark 2.7.

Since op:ΔΔ:opΔΔ{\rm op}:\Delta\to\Delta preserves inert morphisms, the naturality of the unstraightening construction implies that from a lax functor L:𝒜:𝐿𝒜L:\mathcal{A}\rightsquigarrow\mathcal{B} one gets a lax functor Lop:𝒜opop:superscript𝐿opsuperscript𝒜opsuperscriptopL^{\rm op}:\mathcal{A}^{\rm op}\rightsquigarrow\mathcal{B}^{\rm op}.

It is straightforward to extend this notion to define symmetric monoidal lax functors, that is, functors between symmetric monoidal (,2)2(\infty,2)-categories which preserve the symmetric monoidal structure but only laxly preserve composition, as follows.

Definition 2.8.

A symmetric monoidal lax functor L:𝒜:𝐿superscript𝒜tensor-productsuperscripttensor-productL:\mathcal{A}^{\otimes}\rightsquigarrow\mathcal{B}^{\otimes} between symmetric monoidal (,2)2(\infty,2)-categories 𝒜,:in×Δop𝒞at:superscript𝒜tensor-productsuperscripttensor-productsubscriptinsuperscriptdouble-struck-Δop𝒞subscriptat\mathcal{A}^{\otimes},\mathcal{B}^{\otimes}:\mathcal{F}{\rm in}_{*}\times\mathbb{\Delta}^{\rm op}\to\mathcal{C}{\rm at}_{\infty} is a morphism

Un(𝒜)Unsuperscript𝒜tensor-product\textstyle{{\rm Un}\left(\mathcal{A}^{\otimes}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}L𝐿\scriptstyle{L}Un()Unsuperscripttensor-product\textstyle{{\rm Un}\left(\mathcal{B}^{\otimes}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}in×Δopsubscriptinsuperscriptdouble-struck-Δop\textstyle{\mathcal{F}{\rm in}_{*}\times\mathbb{\Delta}^{\rm op}}

such that L𝐿L sends cocartesian lifts of morphisms in in×(Δin)opsubscriptinsuperscriptsubscriptdouble-struck-Δinop\mathcal{F}{\rm in}_{*}\times(\mathbb{\Delta}_{\rm in})^{\rm op} to cocartesian morphisms.

2.3 Corepresenting algebra objects

To specify an algebra object in a symmetric monoidal (,2)2(\infty,2)-category one must provide not just an associative and unital binary operation, but a coherent choice of higher associativity and unitality data. To package together all of this data we will make use of a category originally introduced by Pirashvili [25].

Denote by AlgAlg{\rm Alg} the category having as objects finite sets. A morphism in AlgAlg{\rm Alg} is a function p:XY:𝑝𝑋𝑌p:X\to Y along with a choice of linear ordering of the (possibly empty) preimages p1(y)superscript𝑝1𝑦p^{-1}(y) for each yY𝑦𝑌y\in Y. The composition of a pair of composable morphisms

X0subscript𝑋0\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}p1subscript𝑝1\scriptstyle{p_{1}}X1subscript𝑋1\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}p2subscript𝑝2\scriptstyle{p_{2}}X2subscript𝑋2\textstyle{X_{2}}

is the composition of the underlying functions, with linear ordering on (p2p1)1(x2)superscriptsubscript𝑝2subscript𝑝11subscript𝑥2(p_{2}p_{1})^{-1}(x_{2}) given by

(p2p1)1(x2)=x1p21(x2)p11(x1),superscriptsubscript𝑝2subscript𝑝11subscript𝑥2subscriptsubscript𝑥1superscriptsubscript𝑝21subscript𝑥2superscriptsubscript𝑝11subscript𝑥1(p_{2}p_{1})^{-1}(x_{2})=\sum_{x_{1}\in p_{2}^{-1}(x_{2})}p_{1}^{-1}(x_{1}),

where the sum denotes the monoidal structure on Δ+subscriptΔ\Delta_{+}, the category of finite linear orders, introduced in Eq. 1.4. The disjoint union endows AlgAlg{\rm Alg} with a symmetric monoidal structure.

The category AlgAlg{\rm Alg} corepresents algebra objects in the sense that, for a symmetric monoidal category C𝐶C, symmetric monoidal functors AlgCAlg𝐶{\rm Alg}\to C are the same as algebra objects in C𝐶C. This is because AlgAlg{\rm Alg} is the category of operators [22] for the ΣΣ\Sigma-operad of associative algebras, that is, HomAlg(n¯,1¯)Σnsimilar-to-or-equalssubscriptHomAlg¯𝑛¯1subscriptΣ𝑛{\rm Hom}_{\rm Alg}(\underline{n},\underline{1})\simeq\Sigma_{n} and all morphisms in AlgAlg{\rm Alg} are, up to precomposition with an isomorphism, disjoint unions of these.

Remark 2.9.

One only needs a monoidal structure on a category to define algebra objects in it. The simpler category Δ+subscriptΔ\Delta_{+} corepresents algebra objects in monoidal categories and so one can equivalently define an algebra object in a symmetric monoidal category C𝐶C to be a monoidal functor from Δ+subscriptΔ\Delta_{+} to C𝐶C. However, one needs at least a braiding on a monoidal category to define bialgebra objects in it, and in all the examples which concern us the braiding is in fact symmetric. As this paper lays the foundations for our work on higher categorical bialgebras [24, 23] it is therefore crucial that we make use of AlgAlg{\rm Alg} rather than Δ+subscriptΔ\Delta_{+}.

To corepresent algebra objects in a symmetric monoidal (,2)2(\infty,2)-category it suffices to present AlgAlg{\rm Alg} in the model of symmetric monoidal (,2)2(\infty,2)-categories that we use in this paper. Observe that for 𝒞𝒞\mathcal{C} an \infty-category and F:𝒞1𝒞2:𝐹subscript𝒞1subscript𝒞2F:\mathcal{C}_{1}\to\mathcal{C}_{2} a fully faithful functor, both the right and left Kan extensions,

F,F!:un(𝒞1,𝒞)un(𝒞2,𝒞),:subscript𝐹subscript𝐹unsubscript𝒞1𝒞unsubscript𝒞2𝒞F_{*},F_{!}:\mathcal{F}{\rm un}(\mathcal{C}_{1},\mathcal{C})\to\mathcal{F}{\rm un}(\mathcal{C}_{2},\mathcal{C}),

are fully faithful should they exist.

Definition 2.10.

Let 𝒞𝒞\mathcal{C} be an \infty-category and F:𝒞1𝒞2:𝐹subscript𝒞1subscript𝒞2F:\mathcal{C}_{1}\to\mathcal{C}_{2} be a fully faithful functor. We call functors in the image of Fsubscript𝐹F_{*} cartesian and those in the image of F!subscript𝐹F_{!} cocartesian. We denote these full subcategories, respectively, by uncart(𝒞2,𝒞)superscriptuncartsubscript𝒞2𝒞\mathcal{F}{\rm un}^{\rm cart}(\mathcal{C}_{2},\mathcal{C}) and uncocart(𝒞2,𝒞)superscriptuncocartsubscript𝒞2𝒞\mathcal{F}{\rm un}^{\rm cocart}(\mathcal{C}_{2},\mathcal{C}), with the functor F𝐹F to be understood implicitly from the context.

Remark 2.11.

We follow Haugseng’s terminology [10] as our definition of a (co)cartesian functor is a generalisation of the one given there. In Section 2.2 we discussed cocartesian morphisms and cocartesian fibrations. These are distinct notions from the one being introduced now, but as these terms are used in different contexts there is little fear of confusion.

For each set S𝑆S, denote by Π(S)Π𝑆\Pi({S}) the poset of subsets of S𝑆S ordered by inclusion. These assemble into a functor Π():FinopCat:ΠsuperscriptsubscriptFinopCat\Pi({-}):{\rm Fin}_{*}^{\rm op}\to{\rm Cat} by declaring the image of a pointed map f:ST:𝑓subscript𝑆subscript𝑇f:S_{*}\to T_{*} to be

Π(f):Π(T)Π(S),Uf1(U).:Π𝑓formulae-sequenceΠ𝑇Π𝑆maps-to𝑈superscript𝑓1𝑈\Pi({f}):\Pi({T})\to\Pi({S}),\quad U\mapsto f^{-1}(U).

For each SFinsubscript𝑆subscriptFinS_{*}\in{\rm Fin}_{*} there is a full subcategory P(S)𝑃𝑆\textstyle{P({S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Π(S)Π𝑆\textstyle{\Pi({S})} consisting of the singleton subsets. The \infty-categories 𝒫(S)𝒫𝑆\mathcal{P}({S}) and Π(S)double-struck-Π𝑆\mathbb{\Pi}(S) are, respectively, the nerves of P(S)𝑃𝑆P({S}) and Π(S)Π𝑆\Pi({S}).

Example 2.12.

A functor Π({1,2})op𝒞double-struck-Πsuperscript12op𝒞\mathbb{\Pi}(\{1,2\})^{\rm op}\to\mathcal{C} is a diagram,

c{1,2}subscript𝑐12\textstyle{c_{\{1,2\}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}c{1}subscript𝑐1\textstyle{c_{\{1\}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}c{2}subscript𝑐2\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces c_{\{2\}}}csubscript𝑐\textstyle{c_{\emptyset}}

Such a diagram is cartesian if it presents c{1,2}subscript𝑐12c_{\{1,2\}} as the product of c{1}subscript𝑐1c_{\{1\}} and c{2}subscript𝑐2c_{\{2\}} and csubscript𝑐c_{\emptyset} is terminal. Similarly, a cartesian functor Π(S)op𝒞double-struck-Πsuperscript𝑆op𝒞\mathbb{\Pi}(S)^{\rm op}\to\mathcal{C} encodes a coherent choice of products for a collection of objects of 𝒞𝒞\mathcal{C} labelled by the elements of S𝑆S.

Recall that the symmetric monoidal structure on an \infty-category 𝒟𝒟\mathcal{D} having finite coproducts is given by the functor

𝒟:in𝒞at,Suncocart(Π(S),𝒟).:superscript𝒟coproductformulae-sequencesubscriptin𝒞subscriptatmaps-tosubscript𝑆superscriptuncocartdouble-struck-Π𝑆𝒟\mathcal{D}^{\amalg}:\mathcal{F}{\rm in}_{*}\to\mathcal{C}{\rm at}_{\infty},\quad S_{*}\mapsto\mathcal{F}{\rm un}^{\rm cocart}\left(\mathbb{\Pi}(S),\mathcal{D}\right).

While the disjoint union is not the coproduct in AlgAlg{\rm Alg}, it is the case that given morphisms pi:XiYi:subscript𝑝𝑖subscript𝑋𝑖subscript𝑌𝑖p_{i}:X_{i}\to Y_{i} there is a unique morphism making the following diagram commute

X1subscript𝑋1\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}p1subscript𝑝1\scriptstyle{p_{1}}Y1subscript𝑌1\textstyle{Y_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}X1X2subscript𝑋1coproductsubscript𝑋2\textstyle{X_{1}\coprod X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Y1Y2subscript𝑌1coproductsubscript𝑌2\textstyle{Y_{1}\coprod Y_{2}}X2subscript𝑋2\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}p2subscript𝑝2\scriptstyle{p_{2}}Y2subscript𝑌2\textstyle{Y_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}

Define a functor F:Π(S)×[n]Alg:𝐹Π𝑆delimited-[]𝑛AlgF:\Pi({S})\times[n]\to{\rm Alg} to be cocartesian if for each i[n]𝑖delimited-[]𝑛i\in[n], the composite

Π(S)Π𝑆\textstyle{\Pi({S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}F(,i)𝐹𝑖\scriptstyle{F(-,i)}AlgAlg\textstyle{{\rm Alg}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}forgetforget\scriptstyle{\rm forget}FinFin\textstyle{\rm Fin}

is cocartesian. One has by the above that the functor

𝒜lg:Fin×ΔopqCat0:𝒜superscriptlgcoproductsubscriptFinsuperscriptΔopsuperscriptqCat0\mathcal{A}{\rm lg}^{\amalg}:{\rm Fin}_{*}\times\Delta^{\rm op}\to{\rm qCat}^{0}

sending (S,[n])subscript𝑆delimited-[]𝑛(S_{*},[n]) to the set of cocartesian functors Π(S)×[n]AlgΠ𝑆delimited-[]𝑛Alg\Pi({S})\times[n]\to{\rm Alg} is a symmetric monoidal (,2)2(\infty,2)-category.

We can now define the algebraic structures which are the focus of the remainder of this paper.

Definition 2.13.

Let superscripttensor-product\mathcal{B}^{\otimes} be a symmetric monoidal (,2)2(\infty,2)-category.

  • An algebra object in superscripttensor-product\mathcal{B}^{\otimes} is a symmetric monoidal functor 𝒜lg𝒜superscriptlgcoproductsuperscripttensor-product\mathcal{A}{\rm lg}^{\amalg}\to\mathcal{B}^{\otimes}.

  • A lax algebra object in superscripttensor-product\mathcal{B}^{\otimes} is a symmetric monoidal lax functor 𝒜lg𝒜superscriptlgcoproductsuperscripttensor-product\mathcal{A}{\rm lg}^{\amalg}\rightsquigarrow\mathcal{B}^{\otimes}.

Remark 2.14.

One can readily dualise the preceding discussion to define coalgebra objects in symmetric monoidal (,2)2(\infty,2)-categories. Namely, a (lax) coalgebra object in superscripttensor-product\mathcal{B}^{\otimes} is a symmetric monoidal (lax) functor from 𝒞oalg:=(𝒜lg)opassign𝒞superscriptoalgcoproductsuperscript𝒜superscriptlgcoproductop\mathcal{C}{\rm oalg}^{\amalg}:=(\mathcal{A}{\rm lg}^{\amalg})^{\rm op} to superscripttensor-product\mathcal{B}^{\otimes}.

It is worth taking a moment to informally discuss the exact nature of a lax algebra object A𝐴A, as it is slightly more subtle than one might initially expect. For each string of composable morphisms in AlgAlg{\rm Alg},

n¯0subscript¯𝑛0\textstyle{\underline{n}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}p1subscript𝑝1\scriptstyle{p_{1}}n¯1subscript¯𝑛1\textstyle{\underline{n}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}p2subscript𝑝2\scriptstyle{p_{2}}\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}pksubscript𝑝𝑘\scriptstyle{p_{k}}n¯ksubscript¯𝑛𝑘\textstyle{\underline{n}_{k}}

one has 222-morphisms

An1superscript𝐴tensor-productabsentsubscript𝑛1\textstyle{A^{\otimes n_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}p2subscript𝑝2\scriptstyle{p_{2}}\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}pk1subscript𝑝𝑘1\scriptstyle{p_{k-1}}Ank1superscript𝐴tensor-productabsentsubscript𝑛𝑘1\textstyle{A^{\otimes n_{k-1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}pksubscript𝑝𝑘\scriptstyle{p_{k}}An0superscript𝐴tensor-productabsentsubscript𝑛0\textstyle{A^{\otimes n_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}p1subscript𝑝1\scriptstyle{p_{1}}pkp1subscript𝑝𝑘subscript𝑝1\scriptstyle{p_{k}\circ\cdots\circ p_{1}}Anksuperscript𝐴tensor-productabsentsubscript𝑛𝑘\textstyle{A^{\otimes n_{k}}}

which are compatible with disjoint union and the composition of morphisms in AlgAlg{\rm Alg}. In particular, the witness to the lax associativity of the product on A𝐴A is a diagram

A3superscript𝐴tensor-productabsent3\textstyle{A^{\otimes 3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}μIdtensor-product𝜇Id\scriptstyle{\mu\otimes{\rm Id}}Idμtensor-productId𝜇\scriptstyle{{\rm Id}\otimes\mu}A2superscript𝐴tensor-productabsent2\textstyle{A^{\otimes 2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}μ𝜇\scriptstyle{\mu}A2superscript𝐴tensor-productabsent2\textstyle{A^{\otimes 2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}μ𝜇\scriptstyle{\mu}A.𝐴\textstyle{A\,.}

3 The (,2)2(\infty,2)-category of bispans

The construction which associates the \infty-category 𝒮pan(𝒞)𝒮pan𝒞\mathcal{S}{\rm pan}\left({\mathcal{C}}\right) to an \infty-category 𝒞𝒞\mathcal{C} having finite limits can be iterated to form (,n)𝑛(\infty,n)-categories for each n𝑛n. Informally, a 222-morphism between spans is a ‘span of spans’, that is a diagram of the form

d𝑑\textstyle{d\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}c𝑐\textstyle{c}e𝑒\textstyle{e\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}csuperscript𝑐\textstyle{c^{\prime}}dsuperscript𝑑\textstyle{d^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}

while a 333-morphism is a ‘span of spans of spans’, and so on. The cartesian product in 𝒞𝒞\mathcal{C} endows these (,n)𝑛(\infty,n)-categories with a symmetric monoidal structure. The rigorous construction of these symmetric monoidal (,n)𝑛(\infty,n)-categories has been carried out by Haugseng [10].

Section 3.1 is a brief discussion on the twisted arrow construction, a construction which appears throughout this work. In Section 3.2 we review Haugseng’s construction in the case that concerns us, namely the construction of the symmetric monoidal (,2)2(\infty,2)-category 𝒮pan2×(𝒞)𝒮superscriptsubscriptpan2𝒞\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right) of bispans in 𝒞𝒞\mathcal{C}. In Section 3.3 we prove that 𝒮pan2×(𝒞)𝒮superscriptsubscriptpan2𝒞\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right) is semistrict, a technical condition which simplifies the description of lax functors. Finally, in Section 3.4 we determine explicitly the unstraightening of 𝒮pan2×(N(C))𝒮superscriptsubscriptpan2𝑁𝐶\mathcal{S}{\rm pan}_{2}^{\times}\left({N(C)}\right) for C𝐶C an ordinary category having finite limits.

3.1 The twisted arrow category

The twisted arrow category ([21] IX.6.3), ΣDsuperscriptΣ𝐷\Sigma^{D}, of an ordinary category D𝐷D will be a recurring character in this work. It will first appear in the definition of the (,2)2(\infty,2)-category of bispans in Section 3.2 and its subsequent reappearances will be tied to various constructions involving bispans.

For an ordinary category D𝐷D, let ΣDsuperscriptΣ𝐷\Sigma^{D} be the category having as objects arrows f:dd:𝑓𝑑superscript𝑑f:d\to d^{\prime} in D𝐷D, and morphisms from f1subscript𝑓1f_{1} to f2subscript𝑓2f_{2}

d1subscript𝑑1\textstyle{d_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}f1subscript𝑓1\scriptstyle{f_{1}}d1superscriptsubscript𝑑1\textstyle{d_{1}^{\prime}}d2subscript𝑑2\textstyle{d_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}f2subscript𝑓2\scriptstyle{f_{2}}d2superscriptsubscript𝑑2\textstyle{d_{2}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}

Remembering only the source and target of an object of ΣDsuperscriptΣ𝐷\Sigma^{D} defines a forgetful functor ΣDD×DopsuperscriptΣ𝐷𝐷superscript𝐷op\Sigma^{D}\to D\times D^{\rm op}. Furthermore, the twisted arrow categories assemble into a functor Σ:CatCat:superscriptΣCatCat\Sigma^{-}:{\rm Cat}\to{\rm Cat}.

Remark 3.1.

There are two equally canonical conventions for the definition of the twisted arrow category, the second of which has morphisms given by diagrams

d1subscript𝑑1\textstyle{d_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}f1subscript𝑓1\scriptstyle{f_{1}}d1superscriptsubscript𝑑1\textstyle{d_{1}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}d2subscript𝑑2\textstyle{d_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}f2subscript𝑓2\scriptstyle{f_{2}}d2superscriptsubscript𝑑2\textstyle{d_{2}^{\prime}}

We follow the convention used by Haugseng [10], while the second convention is used by Barwick [2] and Lurie ([18] 5.2.1).

A number of constructions in later sections involve writing explicit functors of the form ΣDCsuperscriptΣ𝐷𝐶\Sigma^{D}\to C for C𝐶C a category having finite limits. It turns out that such functors can be equivalently described as normal oplax functors Dsp(C)𝐷sp𝐶D\nrightarrow{\rm sp}\left(C\right), where sp(C)sp𝐶{\rm sp}\left(C\right) is a bicategory which we shall describe shortly. For our purposes this latter description will often be more convenient.

Given a category C𝐶C having finite limits one can define a bicategory sp(C)sp𝐶{\rm sp}\left(C\right) [3] having the same objects as C𝐶C, 111-morphisms given by spans, and 222-morphisms given by diagrams

d𝑑\textstyle{d\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}c𝑐\textstyle{c}c.superscript𝑐\textstyle{c^{\prime}\,.}dsuperscript𝑑\textstyle{d^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}

Horizontal composition is given by pullbacks, chosen once and for all. The identity 111-morphism for an object c𝑐c is the span c𝑐\textstyle{c\ignorespaces\ignorespaces\ignorespaces\ignorespaces}c𝑐\textstyle{c\ignorespaces\ignorespaces\ignorespaces\ignorespaces}c𝑐\textstyle{c}.

Remark 3.2.

Note that this bicategory is similar to, but distinct from, the (,2)2(\infty,2)-category of bispans in C𝐶C that we shall introduce in Section 3.2. They have different 222-morphisms and, unlike the (,2)2(\infty,2)-category of bispans in C𝐶C, sp(C)sp𝐶{\rm sp}\left(C\right) has no k𝑘k-morphisms for k>2𝑘2k>2.

A normal oplax functor F:Dsp(C):𝐹𝐷sp𝐶F:D\nrightarrow{\rm sp}\left(C\right) consists of, for each object dD𝑑𝐷d\in D an object F(d)C𝐹𝑑𝐶F(d)\in C, for each morphism f:cd:𝑓𝑐𝑑f:c\to d in D𝐷D a span

F(f)𝐹𝑓\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces F(f)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}F(c)𝐹𝑐\textstyle{F(c)}F(d)𝐹𝑑\textstyle{F(d)}

and for each pair of composable morphisms c𝑐\textstyle{c\ignorespaces\ignorespaces\ignorespaces\ignorespaces}f𝑓\scriptstyle{f}d𝑑\textstyle{d\ignorespaces\ignorespaces\ignorespaces\ignorespaces}g𝑔\scriptstyle{g}e𝑒\textstyle{e} a diagram

F(gf)𝐹𝑔𝑓\textstyle{F(g\circ f)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Φg,fsubscriptΦ𝑔𝑓\scriptstyle{\Phi_{g,f}}F(c)𝐹𝑐\textstyle{F(c)}F(e).𝐹𝑒\textstyle{F(e)\,.}F(g)×F(d)F(f)subscript𝐹𝑑𝐹𝑔𝐹𝑓\textstyle{F(g)\times_{F(d)}F(f)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}

For each object dD𝑑𝐷d\in D it must be that F(Idd)=IdF(d)𝐹subscriptId𝑑subscriptId𝐹𝑑F({\rm Id}_{d})={\rm Id}_{F(d)}, for each morphism c𝑐\textstyle{c\ignorespaces\ignorespaces\ignorespaces\ignorespaces}f𝑓\scriptstyle{f}d𝑑\textstyle{d} it must be that ΦIdd,f=IdF(f)=Φf,IdcsubscriptΦsubscriptId𝑑𝑓subscriptId𝐹𝑓subscriptΦ𝑓subscriptId𝑐\Phi_{{\rm Id}_{d},f}={\rm Id}_{F(f)}=\Phi_{f,{\rm Id}_{c}}, and, suppressing associator isomorphisms, for each string of composable morphisms b𝑏\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces}f𝑓\scriptstyle{f}c𝑐\textstyle{c\ignorespaces\ignorespaces\ignorespaces\ignorespaces}g𝑔\scriptstyle{g}d𝑑\textstyle{d\ignorespaces\ignorespaces\ignorespaces\ignorespaces}h\scriptstyle{h}e𝑒\textstyle{e},

(IdF(h)×F(d)Φg,f)Φh,gf=(Φh,g×F(c)IdF(f))Φhg,f,subscript𝐹𝑑subscriptId𝐹subscriptΦ𝑔𝑓subscriptΦ𝑔𝑓subscript𝐹𝑐subscriptΦ𝑔subscriptId𝐹𝑓subscriptΦ𝑔𝑓\left({\rm Id}_{F(h)}\times_{F(d)}\Phi_{g,f}\right)\circ\Phi_{h,gf}=\left(\Phi_{h,g}\times_{F(c)}{\rm Id}_{F(f)}\right)\circ\Phi_{hg,f}, (3.1)

as morphisms F(hgf)F(h)×F(d)F(g)×F(c)F(f)𝐹𝑔𝑓subscript𝐹𝑐subscript𝐹𝑑𝐹𝐹𝑔𝐹𝑓F(hgf)\to F(h)\times_{F(d)}F(g)\times_{F(c)}F(f).

Theorem 3.3 ([6] 3.4.1).

For any category D𝐷D and any category C𝐶C having finite limits there is a natural bijection

Hom(ΣD,C)Homn.oplax(D,sp(C)),similar-to-or-equalsHomsuperscriptΣ𝐷𝐶superscriptHomformulae-sequencenoplax𝐷sp𝐶\mathrm{Hom}\left(\Sigma^{D},C\right)\simeq\mathrm{Hom}^{\rm n.oplax}\left(D,{\rm sp}\left(C\right)\right),

where Homn.oplaxsuperscriptHomformulae-sequencenoplax\mathrm{Hom}^{\rm n.oplax} denotes the set of normal oplax functors.

The isomorphism is given as follows. A normal oplax functor F:Dsp(C):𝐹𝐷sp𝐶F:D\nrightarrow{\rm sp}\left(C\right) associates to each diagram

d1subscript𝑑1\textstyle{d_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}f1subscript𝑓1\scriptstyle{f_{1}}g𝑔\scriptstyle{g}d1superscriptsubscript𝑑1\textstyle{d_{1}^{\prime}}d2subscript𝑑2\textstyle{d_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}f2subscript𝑓2\scriptstyle{f_{2}}d2.superscriptsubscript𝑑2\textstyle{d_{2}^{\prime}\ .\ignorespaces\ignorespaces\ignorespaces\ignorespaces}h\scriptstyle{h}

a diagram

F(f1)𝐹subscript𝑓1\textstyle{F(f_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Fg,hsubscript𝐹𝑔\scriptstyle{F_{g,h}}F(g)𝐹𝑔\textstyle{F(g)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}F(f2)𝐹subscript𝑓2\textstyle{F(f_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}F(h)𝐹\textstyle{F(h)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}F(d1)𝐹subscript𝑑1\textstyle{F(d_{1})}F(d2)𝐹subscript𝑑2\textstyle{F(d_{2})}F(d2)𝐹superscriptsubscript𝑑2\textstyle{F(d_{2}^{\prime})}F(d1).𝐹superscriptsubscript𝑑1\textstyle{F(d_{1}^{\prime})\,.}

The corresponding functor F~:ΣDC:~𝐹superscriptΣ𝐷𝐶\tilde{F}:\Sigma^{D}\to C is then

F~:fF(f),((g,h):f1f2)Fg,h.\tilde{F}:f\mapsto F(f),\quad\left((g,h):f_{1}\to f_{2}\right)\mapsto F_{g,h}.

Finally, an immediate corollary of Theorem 3.3 is the following:

Corollary 3.4.

Let CatlexsuperscriptCatlex{\rm Cat}^{\mathrm{lex}} denote the category of categories having finite limits and finite limit preserving functors between them. Then for any diagram L:JCat:𝐿𝐽CatL:J\to{\rm Cat}, the functors

Hom(ΣcolimL,),Hom(colimjJΣL(j),):CatlexSet:HomsuperscriptΣcolim𝐿Homsubscriptcolim𝑗𝐽superscriptΣ𝐿𝑗superscriptCatlexSet\mathrm{Hom}\left(\Sigma^{\operatornamewithlimits{colim}L},-\right),\mathrm{Hom}\left(\operatornamewithlimits{colim}_{j\in J}\Sigma^{L(j)},-\right):{\rm Cat}^{\mathrm{lex}}\to{\rm Set}

are naturally isomorphic.

Proof.

Since bicategories and normal oplax functors between them assemble into a category, the set-valued functor Homn.oplax(,)superscriptHomformulae-sequencenoplax\mathrm{Hom}^{\rm n.oplax}(-,-) sends colimits in the first variable to limits in sets. Let L:JCat:𝐿𝐽CatL:J\to{\rm Cat} be a diagram in CatCat{\rm Cat}. By Theorem 3.3 one has for each C𝐶C having finite limits the following string of natural bijections

Hom(ΣcolimL,C)HomsuperscriptΣcolim𝐿𝐶\displaystyle\mathrm{Hom}\left(\Sigma^{\operatornamewithlimits{colim}L},C\right) Homn.oplax(colimL,sp(C))limjJHomn.oplax(L(j),sp(C))similar-to-or-equalsabsentsuperscriptHomformulae-sequencenoplaxcolim𝐿sp𝐶similar-to-or-equalssubscriptlim𝑗𝐽superscriptHomformulae-sequencenoplax𝐿𝑗sp𝐶\displaystyle\simeq\mathrm{Hom}^{\rm n.oplax}\left(\operatornamewithlimits{colim}L,{\rm sp}\left(C\right)\right)\simeq\operatornamewithlimits{lim}_{j\in J}\mathrm{Hom}^{\rm n.oplax}\left(L(j),{\rm sp}\left(C\right)\right)
limjJHom(ΣL(j),C)Hom(colimjJΣL(j),C).similar-to-or-equalsabsentsubscriptlim𝑗𝐽HomsuperscriptΣ𝐿𝑗𝐶similar-to-or-equalsHomsubscriptcolim𝑗𝐽superscriptΣ𝐿𝑗𝐶\displaystyle\simeq\operatornamewithlimits{lim}_{j\in J}\mathrm{Hom}\left(\Sigma^{L(j)},C\right)\simeq\mathrm{Hom}\left(\operatornamewithlimits{colim}_{j\in J}\Sigma^{L(j)},C\right).

3.2 Definition of 𝒮pan2×(𝒞)𝒮superscriptsubscriptpan2𝒞\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right)

Haugseng’s construction of the symmetric monoidal (,2)2(\infty,2)-category 𝒮pan2×(𝒞)𝒮superscriptsubscriptpan2𝒞\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right) is an iteration of Barwick’s construction of the \infty-category 𝒮pan(𝒞)𝒮pan𝒞\mathcal{S}{\rm pan}\left({\mathcal{C}}\right) [2]. One first defines a symmetric monoidal double \infty-category 𝒮pan2×(𝒞)¯𝒮eg(𝒞at)¯𝒮superscriptsubscriptpan2𝒞𝒮eg𝒞superscriptsubscriptattensor-product\overline{\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right)}\in\mathcal{S}{\rm eg}\left({\mathcal{C}{\rm at}_{\infty}^{\otimes}}\right) which fails to be in 𝒮eg0(𝒞at)𝒮subscripteg0𝒞superscriptsubscriptattensor-product\mathcal{S}{\rm eg}_{0}\left({\mathcal{C}{\rm at}_{\infty}^{\otimes}}\right). One then remedies this problem by defining 𝒮pan2×(𝒞):=𝒰(𝒮pan2×(𝒞)¯)assign𝒮superscriptsubscriptpan2𝒞𝒰¯𝒮superscriptsubscriptpan2𝒞\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right):=\mathcal{U}\left(\overline{\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right)}\right).

The morphisms in the (,2)2(\infty,2)-category of bispans are given by diagrams in 𝒞𝒞\mathcal{C} of the following form. Let Σn:=Σ[n]assignsuperscriptΣ𝑛superscriptΣdelimited-[]𝑛\Sigma^{n}:=\Sigma^{[n]} be the twisted arrow construction of [n]delimited-[]𝑛[n]. Explicitly, this is the opposite of the poset of non-empty intervals in [n]delimited-[]𝑛[n]. Similarly, the category Σn,k:=Σ[n]×[k]=Σn×ΣkassignsuperscriptΣ𝑛𝑘superscriptΣdelimited-[]𝑛delimited-[]𝑘superscriptΣ𝑛superscriptΣ𝑘\Sigma^{n,k}:=\Sigma^{[n]\times[k]}=\Sigma^{n}\times\Sigma^{k} is opposite of the poset of non-empty rectangles in [n]×[k]delimited-[]𝑛delimited-[]𝑘[n]\times[k]. These assemble into a functor

Σ,:(Δ)2Cat.:superscriptΣsuperscriptΔ2Cat\Sigma^{\bullet,\bullet}:(\Delta)^{2}\to{\rm Cat}.

There is a full subcategory ΛnsuperscriptΛ𝑛\textstyle{\Lambda^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}ΣnsuperscriptΣ𝑛\textstyle{\Sigma^{n}} consisting of intervals [i;j]𝑖𝑗[i;j] with |ji|1𝑗𝑖1|j-i|\leq 1 and hence a full subcategory Λn,ksuperscriptΛ𝑛𝑘\textstyle{\Lambda^{n,k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Σn,ksuperscriptΣ𝑛𝑘\textstyle{\Sigma^{n,k}}. The \infty-categories Λnsuperscriptdouble-struck-Λ𝑛\mathbb{\Lambda}^{n} and Σnsuperscriptdouble-struck-Σ𝑛\mathbb{\Sigma}^{n} are, respectively, the nerves of ΛnsuperscriptΛ𝑛\Lambda^{n} and ΣnsuperscriptΣ𝑛\Sigma^{n}.

Example 3.5.

A functor Σ2𝒞superscriptdouble-struck-Σ2𝒞\mathbb{\Sigma}^{2}\to\mathcal{C} is a diagram,

c[0;2]subscript𝑐02\textstyle{c_{[0;2]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}c[0;1]subscript𝑐01\textstyle{c_{[0;1]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}c[1;2]subscript𝑐12\textstyle{c_{[1;2]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}c[0;0]subscript𝑐00\textstyle{c_{[0;0]}}c[1;1]subscript𝑐11\textstyle{c_{[1;1]}}c[2;2].subscript𝑐22\textstyle{c_{[2;2]}\,.}

Such a diagram is cartesian if it is the right Kan extension of its restriction to Λ2superscriptdouble-struck-Λ2\mathbb{\Lambda}^{2}, i.e., if the middle square is a pullback in 𝒞𝒞\mathcal{C}. In general, a functor Σn𝒞superscriptdouble-struck-Σ𝑛𝒞\mathbb{\Sigma}^{n}\to\mathcal{C} is pyramid of spans on n+1𝑛1n+1 objects. It being cartesian says higher tiers of this pyramid consist of a coherent choice of pullbacks of the n𝑛n spans along the bottom two tiers.

Presenting 𝒞at𝒞subscriptat\mathcal{C}{\rm at}_{\infty} as 𝒞SS𝒞SS\mathcal{C}{\rm SS} we can now define the symmetric monoidal (,2)2(\infty,2)-category of bispans. For an \infty-category 𝒞𝒞\mathcal{C} having finite limits, consider the functor

Fin×(Δop)2Kan,(S,[n],[k])apcart(Π(S)op×Σn,k,𝒞),formulae-sequencesubscriptFinsuperscriptsuperscriptΔop2Kanmaps-tosubscript𝑆delimited-[]𝑛delimited-[]𝑘superscriptapcartdouble-struck-Πsuperscript𝑆opsuperscriptdouble-struck-Σ𝑛𝑘𝒞{\rm Fin}_{*}\times\left(\Delta^{\rm op}\right)^{2}\to{\rm Kan},\quad(S_{*},[n],[k])\mapsto\mathcal{M}{\rm ap}^{\rm cart}\left(\mathbb{\Pi}(S)^{\rm op}\times\mathbb{\Sigma}^{n,k},\mathcal{C}\right),

which is well-defined by Proposition 3.8 of [10]. Taking the coherent nerve defines a functor 𝒮pan2×(𝒞)¯:in×(Δop)2𝒮:¯𝒮superscriptsubscriptpan2𝒞subscriptinsuperscriptsuperscriptdouble-struck-Δop2𝒮\overline{\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right)}:\mathcal{F}{\rm in}_{*}\times\left(\mathbb{\Delta}^{\rm op}\right)^{2}\to\mathcal{S}.

Proposition 3.6.

Let 𝒞𝒞\mathcal{C} be an \infty-category with finite limits. Then the functor

𝒮pan2×(𝒞):=𝒰(𝒮pan2×(𝒞)¯):in×Δop𝒞at:assign𝒮superscriptsubscriptpan2𝒞𝒰¯𝒮superscriptsubscriptpan2𝒞subscriptinsuperscriptdouble-struck-Δop𝒞subscriptat\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right):=\mathcal{U}\left(\overline{\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right)}\right):\mathcal{F}{\rm in}_{*}\times\mathbb{\Delta}^{\rm op}\to\mathcal{C}{\rm at}_{\infty}

is a symmetric monoidal (,2)2(\infty,2)-category. We call this the symmetric monoidal (,2)2(\infty,2)-category of bispans in 𝒞𝒞\mathcal{C}.

Remark 3.7.

Our description of the symmetric monoidal structure arising from the cartesian product on 𝒞𝒞\mathcal{C} differs from Haugseng. He instead presents the symmetric monoidal structure by giving a sequence of (,2+k)2𝑘(\infty,2+k)-categories delooping the (,2)2(\infty,2)-category of bispans ([10] 9.1).

Proof.

By Corollary 6.5 of [10], for each fixed Sinsubscript𝑆subscriptinS_{*}\in\mathcal{F}{\rm in}_{*}, the simplicial \infty-category 𝒮pan2×(𝒞)(S,,)𝒮superscriptsubscriptpan2𝒞subscript𝑆\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right)(S_{*},\bullet,\bullet) is a (,2)2(\infty,2)-category. Consider the following commutative diagram in 𝒮𝒮\mathcal{S},

𝒮pan2×(𝒞)¯(S,[n],[k])¯𝒮superscriptsubscriptpan2𝒞subscript𝑆delimited-[]𝑛delimited-[]𝑘\textstyle{\overline{\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right)}(S_{*},[n],[k])\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\scriptstyle{\wr}sS𝒮pan2×(𝒞)¯({s},[n],[k])subscriptproduct𝑠𝑆¯𝒮superscriptsubscriptpan2𝒞subscript𝑠delimited-[]𝑛delimited-[]𝑘\textstyle{\prod_{s\in S}\overline{\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right)}(\{s\}_{*},[n],[k])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\scriptstyle{\wr}apcart(Π(S)op×Σn,k,𝒞)superscriptapcartdouble-struck-Πsuperscript𝑆opsuperscriptdouble-struck-Σ𝑛𝑘𝒞\textstyle{\mathcal{M}{\rm ap}^{\rm cart}\left(\mathbb{\Pi}(S)^{\rm op}\times\mathbb{\Sigma}^{n,k},\mathcal{C}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}jsuperscript𝑗\scriptstyle{j^{*}}ap(𝒫(S)op×Λn,k,𝒞),ap𝒫superscript𝑆opsuperscriptdouble-struck-Λ𝑛𝑘𝒞\textstyle{\mathcal{M}{\rm ap}\left(\mathcal{P}({S})^{\rm op}\times\mathbb{\Lambda}^{n,k},\mathcal{C}\right)\,,}

where j𝑗j denotes the fully faithful functor including 𝒫(S)op×Λn,k𝒫superscript𝑆opsuperscriptdouble-struck-Λ𝑛𝑘\mathcal{P}({S})^{\rm op}\times\mathbb{\Lambda}^{n,k} into Π(S)op×Σn,kdouble-struck-Πsuperscript𝑆opsuperscriptdouble-struck-Σ𝑛𝑘\mathbb{\Pi}(S)^{\rm op}\times\mathbb{\Sigma}^{n,k}. Since 𝒰𝒰\mathcal{U} preserves limits it suffices to show that the top map is an equivalence.

By Definition 2.10, the category uncart(Π(S)op×Σn,k,𝒞)superscriptuncartdouble-struck-Πsuperscript𝑆opsuperscriptdouble-struck-Σ𝑛𝑘𝒞\mathcal{F}{\rm un}^{\rm cart}\left(\mathbb{\Pi}(S)^{\rm op}\times\mathbb{\Sigma}^{n,k},\mathcal{C}\right) is the image of un(𝒫(S)op×Λn,k,𝒞)un𝒫superscript𝑆opsuperscriptdouble-struck-Λ𝑛𝑘𝒞\mathcal{F}{\rm un}\left(\mathcal{P}({S})^{\rm op}\times\mathbb{\Lambda}^{n,k},\mathcal{C}\right) under jsubscript𝑗j_{*}, the right adjoint of jsuperscript𝑗j^{*}. Since jsubscript𝑗j_{*} is fully faithful it is an equivalence onto its image with inverse jsuperscript𝑗j^{*}. It follows that the top map in the above diagram is an equivalence, as desired. ∎

3.3 𝒮pan2×(𝒞)𝒮superscriptsubscriptpan2𝒞\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right) is semistrict

Describing symmetric monoidal lax functors for general symmetric monoidal (,2)2(\infty,2)-categories can be quite complicated due to the use of the unstraightening construction. For the following class of symmetric monoidal (,2)2(\infty,2)-categories it simplifies greatly.

Recall that a functor N(C)𝒞at𝑁𝐶𝒞subscriptatN(C)\to\mathcal{C}{\rm at}_{\infty} is equivalent to a functor N(C)qCat𝑁𝐶qCat\mathfrak{C}N(C)\to{\rm qCat} of simplicially enriched categories, where \mathfrak{C} is the left adjoint of the coherent nerve ([19] 1.1.5).

Definition 3.8.

For an ordinary category C𝐶C, a functor N(C)𝒞at𝑁𝐶𝒞subscriptatN(C)\to\mathcal{C}{\rm at}_{\infty} is called semistrict if there is a functor of ordinary categories CqCat0𝐶superscriptqCat0C\to{\rm qCat}^{0}, for qCat0superscriptqCat0{\rm qCat}^{0} the ordinary category of quasi-categories and functors between them, such that the following commutes

N(C)𝑁𝐶\textstyle{\mathfrak{C}N(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}ϵitalic-ϵ\scriptstyle{\epsilon}qCatqCat\textstyle{\rm qCat}C𝐶\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}qCat0,superscriptqCat0\textstyle{{\rm qCat}^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\,,}

where ϵitalic-ϵ\epsilon is the counit of the adjunction Ndoes-not-prove𝑁\mathfrak{C}\dashv N. In particular, a symmetric monoidal (,2)2(\infty,2)-category is semistrict if it is semistrict as a functor in×Δop𝒞atsubscriptinsuperscriptdouble-struck-Δop𝒞subscriptat\mathcal{F}{\rm in}_{*}\times\mathbb{\Delta}^{\rm op}\to\mathcal{C}{\rm at}_{\infty}.

The aim of this section is to show that 𝒮pan2×(𝒞)𝒮superscriptsubscriptpan2𝒞\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right) is semistrict. To that end, we must first compute the pullback of \infty-categories in diagram 2.1, or equivalently, the homotopy pullback of

𝒮pan2×(𝒞)¯(S,[n],)¯𝒮superscriptsubscriptpan2𝒞subscript𝑆delimited-[]𝑛\textstyle{\overline{\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right)}(S_{*},[n],\bullet)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}v𝑣\scriptstyle{v}((𝒮pan2×(𝒞)¯(S,[0],)))n+1superscriptsuperscript¯𝒮superscriptsubscriptpan2𝒞subscript𝑆delimited-[]0similar-to-or-equals𝑛1\textstyle{\left(\left(\overline{\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right)}(S_{*},[0],\bullet)\right)^{\simeq}\right)^{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}(𝒮pan2×(𝒞)¯(S,[0],))n+1superscript¯𝒮superscriptsubscriptpan2𝒞subscript𝑆delimited-[]0𝑛1\textstyle{\left(\overline{\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right)}(S_{*},[0],\bullet)\right)^{n+1}} (3.2)

in css, the category of bisimplicial sets endowed with the Rezk model structure [26].

The computation is rendered trivial by the following lemma.

Lemma 3.9.

For each (S,[n])Fin×Δopsubscript𝑆delimited-[]𝑛subscriptFinsuperscriptΔop(S_{*},[n])\in{\rm Fin}_{*}\times\Delta^{\rm op}, the morphism

v:𝒮pan2×(𝒞)¯(S,[n],)(𝒮pan2×(𝒞)¯(S,[0],))n+1:𝑣¯𝒮superscriptsubscriptpan2𝒞subscript𝑆delimited-[]𝑛superscript¯𝒮superscriptsubscriptpan2𝒞subscript𝑆delimited-[]0𝑛1v:\overline{\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right)}(S_{*},[n],\bullet)\to\left(\overline{\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right)}(S_{*},[0],\bullet)\right)^{n+1}

is a fibration in css.

Proof.

It suffices to show that v𝑣v is a Reedy fibration since css is a left Bousfield localisation of the Reedy model structure on the category of bisimplicial sets, and both the source and the target of v𝑣v are fibrant in css ([12] 3.3.16).

For the purposes of this proof we will use the shorthand (,):=ap(,)assignap\mathcal{M}(-,-):=\mathcal{M}{\rm ap}(-,-). Consider the following commutative diagram in SetΔsubscriptSetΔ{\rm Set}_{\Delta},

cart(Π(S)op×Σn,k,𝒞)superscriptcartdouble-struck-Πsuperscript𝑆opsuperscriptdouble-struck-Σ𝑛𝑘𝒞\textstyle{\mathcal{M}^{\rm cart}\left(\mathbb{\Pi}(S)^{\rm op}\times\mathbb{\Sigma}^{n,k},\mathcal{C}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}(cart(Π(S)op×Σn,,𝒞))Δ[k]superscriptsuperscriptcartdouble-struck-Πsuperscript𝑆opsuperscriptdouble-struck-Σ𝑛𝒞Δdelimited-[]𝑘\textstyle{\left(\mathcal{M}^{\rm cart}\left(\mathbb{\Pi}(S)^{\rm op}\times\mathbb{\Sigma}^{n,\bullet},\mathcal{C}\right)\right)^{\partial\Delta\left[{k}\right]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}(cart(Π(S)op×Σ0,k,𝒞))n+1superscriptsuperscriptcartdouble-struck-Πsuperscript𝑆opsuperscriptdouble-struck-Σ0𝑘𝒞𝑛1\textstyle{\left(\mathcal{M}^{\rm cart}\left(\mathbb{\Pi}(S)^{\rm op}\times\mathbb{\Sigma}^{0,k},\mathcal{C}\right)\right)^{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}((cart(Π(S)op×Σ0,,𝒞))n+1)Δ[k]superscriptsuperscriptsuperscriptcartdouble-struck-Πsuperscript𝑆opsuperscriptdouble-struck-Σ0𝒞𝑛1Δdelimited-[]𝑘\textstyle{\left(\left(\mathcal{M}^{\rm cart}\left(\mathbb{\Pi}(S)^{\rm op}\times\mathbb{\Sigma}^{0,\bullet},\mathcal{C}\right)\right)^{n+1}\right)^{\partial\Delta\left[{k}\right]}} (3.3)

The morphism v𝑣v is a Reedy fibration if and only if the induced map ν¯¯𝜈\overline{\nu} from cart(Π(S)op×Σn,k,𝒞)superscriptcartdouble-struck-Πsuperscript𝑆opsuperscriptdouble-struck-Σ𝑛𝑘𝒞\mathcal{M}^{\rm cart}\left(\mathbb{\Pi}(S)^{\rm op}\times\mathbb{\Sigma}^{n,k},\mathcal{C}\right) to the pullback is a Kan fibration of simplicial sets ([26] 2.4).

Let B(n,k)𝐵𝑛𝑘B(n,k) be the pushout of simplicial sets

(Δ[0])n+1×Δ[k]superscriptΔdelimited-[]0coproductabsent𝑛1Δdelimited-[]𝑘\textstyle{(\Delta\left[{0}\right])^{\amalg n+1}\times\partial\Delta\left[{k}\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Δ[n]×Δ[k]Δdelimited-[]𝑛Δdelimited-[]𝑘\textstyle{\Delta\left[{n}\right]\times\partial\Delta\left[{k}\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}(Δ[0])n+1×Δ[k]superscriptΔdelimited-[]0coproductabsent𝑛1Δdelimited-[]𝑘\textstyle{(\Delta\left[{0}\right])^{\amalg n+1}\times\Delta\left[{k}\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}B(n,k).𝐵𝑛𝑘\textstyle{B(n,k)\,.}

The pullback of diagram 3.3 is limΔ[m]B(n,k)cart(Π(S)op×Σm,𝒞)subscriptlimΔdelimited-[]𝑚𝐵𝑛𝑘superscriptcartdouble-struck-Πsuperscript𝑆opsuperscriptdouble-struck-Σ𝑚𝒞\operatornamewithlimits{lim}_{\Delta\left[{m}\right]\to B(n,k)}\mathcal{M}^{\rm cart}\left(\mathbb{\Pi}(S)^{\rm op}\times\mathbb{\Sigma}^{m},\mathcal{C}\right) since (,𝒞):SetΔSetΔ:𝒞subscriptSetΔsubscriptSetΔ\mathcal{M}(-,\mathcal{C}):{\rm Set}_{\Delta}\to{\rm Set}_{\Delta} sends colimits to limits. To prove that ν¯¯𝜈\overline{\nu} is a Kan fibration it suffices to show that

ν:(Π(S)op×Σn,k,𝒞)limΔ[m]B(n,k)(Π(S)op×Σm,𝒞):𝜈double-struck-Πsuperscript𝑆opsuperscriptdouble-struck-Σ𝑛𝑘𝒞subscriptlimΔdelimited-[]𝑚𝐵𝑛𝑘double-struck-Πsuperscript𝑆opsuperscriptdouble-struck-Σ𝑚𝒞\nu:\mathcal{M}\left(\mathbb{\Pi}(S)^{\rm op}\times\mathbb{\Sigma}^{n,k},\mathcal{C}\right)\to\operatornamewithlimits{lim}_{\Delta\left[{m}\right]\to B(n,k)}\mathcal{M}\left(\mathbb{\Pi}(S)^{\rm op}\times\mathbb{\Sigma}^{m},\mathcal{C}\right)

is a Kan fibration as the target of ν¯¯𝜈\overline{\nu} is a union of connected components of the target of ν𝜈\nu.

By left Kan extension along the Yoneda embedding one can extend Σsuperscriptdouble-struck-Σ\mathbb{\Sigma}^{\bullet} to a functor Σ:SetΔSetΔ:superscriptdouble-struck-ΣsubscriptSetΔsubscriptSetΔ\mathbb{\Sigma}^{\bullet}:{\rm Set}_{\Delta}\to{\rm Set}_{\Delta}. Let ϵ:ΔΔ:italic-ϵΔΔ\epsilon:\Delta\to\Delta to be the edgewise subdivison functor sending [n]delimited-[]𝑛[n] to [n][n]opdelimited-[]𝑛superscriptdelimited-[]𝑛op[n]\star[n]^{\rm op} ([2] 2.5). Since Σn=ϵΔ[n]superscriptdouble-struck-Σ𝑛superscriptitalic-ϵΔdelimited-[]𝑛\mathbb{\Sigma}^{n}=\epsilon^{*}\Delta\left[{n}\right] and both functors preserve colimits one has that Σ=ϵsuperscriptdouble-struck-Σsuperscriptitalic-ϵ\mathbb{\Sigma}^{\bullet}=\epsilon^{*}. From this we conclude that Σsuperscriptdouble-struck-Σ\mathbb{\Sigma}^{\bullet} preserves products and monomorphisms.

It follows, therefore, that the target of ν𝜈\nu is (Π(S)op×ΣB(n,k),𝒞)double-struck-Πsuperscript𝑆opsuperscriptdouble-struck-Σ𝐵𝑛𝑘𝒞\mathcal{M}\left(\mathbb{\Pi}(S)^{\rm op}\times\mathbb{\Sigma}^{B(n,k)},\mathcal{C}\right). The morphism ν𝜈\nu is induced by the functor ΣB(n,k)Σn,ksuperscriptdouble-struck-Σ𝐵𝑛𝑘superscriptdouble-struck-Σ𝑛𝑘\mathbb{\Sigma}^{B(n,k)}\to\mathbb{\Sigma}^{n,k}, which is itself induced by the inclusion B(n,k)𝐵𝑛𝑘\textstyle{B(n,k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Δ[n]×Δ[k]Δdelimited-[]𝑛Δdelimited-[]𝑘\textstyle{\Delta\left[{n}\right]\times\Delta\left[{k}\right]} and so is a monomorphism. Hence, by Lemma 3.1.3.6 of [19], ν𝜈\nu is a Kan fibration. ∎

Next, denote by Vn,ksubscript𝑉𝑛𝑘V_{n,k} the sub-poset of [n]×[k]delimited-[]𝑛delimited-[]𝑘[n]\times[k] on those morphisms of the form (Id,g)Id𝑔({\rm Id},g). We say a functor Π(S)op×Σn,k𝒞double-struck-Πsuperscript𝑆opsuperscriptdouble-struck-Σ𝑛𝑘𝒞\mathbb{\Pi}(S)^{\rm op}\times\mathbb{\Sigma}^{n,k}\to\mathcal{C} is vertically constant if morphisms of the form (Id,v)Id𝑣({\rm Id},v) for vΣVn,k𝑣superscriptdouble-struck-Σsubscript𝑉𝑛𝑘v\in\mathbb{\Sigma}^{V_{n,k}} are sent to equivalences.

Definition 3.10.

Let 𝒞𝒞\mathcal{C} be an \infty-category with finite limits and (S,[n])Fin×Δopsubscript𝑆delimited-[]𝑛subscriptFinsuperscriptΔop(S_{*},[n])\in{\rm Fin}_{*}\times\Delta^{\rm op}. Define 𝒮pS,n(𝒞)𝒮subscriptp𝑆𝑛𝒞\mathcal{S}{\rm p}_{S,n}(\mathcal{C}) to be the simplicial set having k𝑘k-simplices the cartesian, vertically constant functors Π(S)op×Σn,k𝒞double-struck-Πsuperscript𝑆opsuperscriptdouble-struck-Σ𝑛𝑘𝒞\mathbb{\Pi}(S)^{\rm op}\times\mathbb{\Sigma}^{n,k}\to\mathcal{C}.

Proposition 3.11.

Let 𝒞𝒞\mathcal{C} be an \infty-category with finite limits. Then for each (S,[n])subscript𝑆delimited-[]𝑛(S_{*},[n]) in Fin×ΔopsubscriptFinsuperscriptΔop{\rm Fin}_{*}\times\Delta^{\rm op}, the simplicial set 𝒮pS,n(𝒞)𝒮subscriptp𝑆𝑛𝒞\mathcal{S}{\rm p}_{S,n}(\mathcal{C}) is a quasi-category. Furthermore, the symmetric monoidal (,2)2(\infty,2)-category of bispans 𝒮pan2×(𝒞)𝒮superscriptsubscriptpan2𝒞\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right) is given by the functor

Fin×ΔopqCat0,(S,[n])𝒮pS,n(𝒞),formulae-sequencesubscriptFinsuperscriptΔopsuperscriptqCat0maps-tosubscript𝑆delimited-[]𝑛𝒮subscriptp𝑆𝑛𝒞{\rm Fin}_{*}\times\Delta^{\rm op}\to{\rm qCat}^{0},\quad(S_{*},[n])\mapsto\mathcal{S}{\rm p}_{S,n}(\mathcal{C}),

and so is semistrict.

Proof.

Recall that for a complete Segal space 𝒳𝒳\mathcal{X} each vertex of 𝒳ksubscript𝒳𝑘\mathcal{X}_{k} determines a sequence of composable morphisms [fi]delimited-[]subscript𝑓𝑖[f_{i}] in the homotopy category of 𝒳𝒳\mathcal{X}. Then 𝒳ksubscriptsuperscript𝒳similar-to-or-equals𝑘\mathcal{X}^{\simeq}_{k} is the full sub simplicial set of 𝒳ksubscript𝒳𝑘\mathcal{X}_{k} on those vertices having each [fi]delimited-[]subscript𝑓𝑖[f_{i}] invertible in the homotopy category ([20] 1.1.11).

In particular, (𝒮pan2×(𝒞)¯(S,[0],))ksubscriptsuperscript¯𝒮superscriptsubscriptpan2𝒞subscript𝑆delimited-[]0similar-to-or-equals𝑘\left(\overline{\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right)}(S_{*},[0],\bullet)\right)^{\simeq}_{k} is the full sub simplicial set of apcart(Π(S)op×Σk,𝒞)superscriptapcartdouble-struck-Πsuperscript𝑆opsuperscriptdouble-struck-Σ𝑘𝒞\mathcal{M}{\rm ap}^{\rm cart}(\mathbb{\Pi}(S)^{\rm op}\times\mathbb{\Sigma}^{k},\mathcal{C}) generated by those functors sending morphisms of the form (Id,f)Id𝑓({\rm Id},f) to equivalences. Since all of the objects in diagram 3.2 are fibrant, Lemma 3.9 implies that the ordinary pullback in css is a complete Segal space presenting the homotopy pullback ([19] A.2.4.4). We can therefore conclude that 𝒮pan2×(𝒞)(S,[n],[k])𝒮superscriptsubscriptpan2𝒞subscript𝑆delimited-[]𝑛delimited-[]𝑘\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right)(S_{*},[n],[k]) is the full sub simplicial set of apcart(Π(S)op×Σn,k,𝒞)superscriptapcartdouble-struck-Πsuperscript𝑆opsuperscriptdouble-struck-Σ𝑛𝑘𝒞\mathcal{M}{\rm ap}^{\rm cart}(\mathbb{\Pi}(S)^{\rm op}\times\mathbb{\Sigma}^{n,k},\mathcal{C}) generated by those functors sending morphisms of the form (Id,v)Id𝑣({\rm Id},v) for vΣVn,k𝑣superscriptdouble-struck-Σsubscript𝑉𝑛𝑘v\in\mathbb{\Sigma}^{V_{n,k}} to equivalences.

The canonical equivalence 𝒞SS𝒞atsimilar-to-or-equals𝒞SS𝒞subscriptat\mathcal{C}{\rm SS}\simeq\mathcal{C}{\rm at}_{\infty} is presented by a Quillen equivalence ([15] 4.11)

css(p1)(SetΔ)Joyal(i1),(p1):X,X,0.:cssfragmentsfragments(subscript𝑝1)subscriptsubscriptSetΔJoyalsuperscriptsubscript𝑖1superscriptsubscript𝑝1maps-tosubscript𝑋subscript𝑋0\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 7.16666pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-7.16666pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{$\textstyle{{\textsc{css}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 7.93349pt\raise-10.80554pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{(p_{1})*}$}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 15.16666pt\raise-4.30554pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 29.16666pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 29.16666pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{$\textstyle{{({\rm Set}_{\Delta})_{\rm Joyal}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 9.23865pt\raise 11.27083pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.21529pt\hbox{$\scriptstyle{(i_{1})^{*}}$}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 15.16667pt\raise 4.30554pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern 3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 7.16667pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\kern 18.16666pt\raise-1.7222pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{\hbox{}}$}}}}}\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\kern 18.16666pt\raise 1.72218pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{\hbox{}}$}}}}}\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 18.16666pt\raise-1.7222pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@stopper}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,\quad(p_{1})^{*}:X_{\bullet,\bullet}\mapsto X_{\bullet,0}\ .

By definition 𝒮pS,n(𝒞)=(p1)𝒮pan2×(𝒞)(S,[n],)𝒮subscriptp𝑆𝑛𝒞superscriptsubscript𝑝1𝒮superscriptsubscriptpan2𝒞subscript𝑆delimited-[]𝑛\mathcal{S}{\rm p}_{S,n}(\mathcal{C})=(p_{1})^{*}\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right)(S_{*},[n],\bullet), which is a quasi-category since (p1)superscriptsubscript𝑝1(p_{1})^{*} preserves fibrations. ∎

Corollary 3.12.

For any \infty-category 𝒞𝒞\mathcal{C} having finite limits its symmetric monoidal (,2)2(\infty,2)-category of bispans is equivalent to its opposite.

Proof.

By Proposition 3.11, the opposite of the symmetric monoidal (,2)2(\infty,2)-category of bispans is

Fin×ΔopqCat0,(S,[n])𝒮pS,[n]op(𝒞),formulae-sequencesubscriptFinsuperscriptΔopsuperscriptqCat0maps-tosubscript𝑆delimited-[]𝑛𝒮subscriptp𝑆superscriptdelimited-[]𝑛op𝒞{\rm Fin}_{*}\times\Delta^{\rm op}\to{\rm qCat}^{0},\quad(S_{*},[n])\mapsto\mathcal{S}{\rm p}_{S,[n]^{\rm op}}(\mathcal{C}),

where the quasi-category 𝒮pS,[n]op(𝒞)𝒮subscriptp𝑆superscriptdelimited-[]𝑛op𝒞\mathcal{S}{\rm p}_{S,[n]^{\rm op}}(\mathcal{C}) has k𝑘k-simplices the cartesian and vertically constant functors Π(S)op×Σ[n]op,k𝒞double-struck-Πsuperscript𝑆opsuperscriptdouble-struck-Σsuperscriptdelimited-[]𝑛op𝑘𝒞\mathbb{\Pi}(S)^{\rm op}\times\mathbb{\Sigma}^{[n]^{\rm op},k}\to\mathcal{C}. Since [n]delimited-[]𝑛[n] is canonically isomorphic to [n]opsuperscriptdelimited-[]𝑛op[n]^{\rm op}, one has an equivalence Σ[n]op,kΣn,ksimilar-to-or-equalssuperscriptdouble-struck-Σsuperscriptdelimited-[]𝑛op𝑘superscriptdouble-struck-Σ𝑛𝑘\mathbb{\Sigma}^{[n]^{\rm op},k}\simeq\mathbb{\Sigma}^{n,k} and hence 𝒮pS,[n]op(𝒞)𝒮pS,n(𝒞)similar-to-or-equals𝒮subscriptp𝑆superscriptdelimited-[]𝑛op𝒞𝒮subscriptp𝑆𝑛𝒞\mathcal{S}{\rm p}_{S,[n]^{\rm op}}(\mathcal{C})\simeq\mathcal{S}{\rm p}_{S,n}(\mathcal{C}). ∎

3.4 Unstraightening 𝒮pan2×(N(C))𝒮superscriptsubscriptpan2𝑁𝐶\mathcal{S}{\rm pan}_{2}^{\times}\left({N(C)}\right)

The aim of this section is to give a description, sufficient for our purposes, of the unstraightening of the symmetric monoidal (,2)2(\infty,2)-category of bispans for an ordinary category C𝐶C having finite limits. In general, determining the quasi-category Un(F)Un𝐹{\rm Un}(F) unstraightening a functor F:𝒟𝒞at:𝐹𝒟𝒞subscriptatF:\mathcal{D}\to\mathcal{C}{\rm at}_{\infty} can be quite difficult. This can, however, be done in the special case when 𝒟=N(D)𝒟𝑁𝐷\mathcal{D}=N(D) for an ordinary category D𝐷D and F𝐹F is semistrict ([19] 3.2.5.2). In this case, the set of k𝑘k-simplices Un(F)kUnsubscript𝐹𝑘{\rm Un}(F)_{k} is the set of pairs

(σNk(D),{τ(J):JF(σ(J¯))}J[k]),𝜎subscript𝑁𝑘𝐷subscriptconditional-set𝜏𝐽𝐽𝐹𝜎¯𝐽𝐽delimited-[]𝑘\left(\sigma\in N_{k}(D),\ \left\{\tau(J):J\to F(\sigma(\overline{J}))\right\}_{\emptyset\neq J\subset[k]}\right), (3.4)

where J¯¯𝐽\overline{J} is the maximal element of J𝐽J. The family of functors τ𝜏\tau must be such that for each JL[k]𝐽𝐿delimited-[]𝑘\emptyset\neq J\subset L\subset[k] the following diagram commutes

J𝐽\textstyle{J\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}τ(J)𝜏𝐽\scriptstyle{\tau(J)}F(σ(J¯))𝐹𝜎¯𝐽\textstyle{F(\sigma(\overline{J}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}F(σ(J¯L¯)\scriptstyle{F(\sigma(\overline{J}\leq\overline{L})}L𝐿\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}τ(L)𝜏𝐿\scriptstyle{\tau(L)}F(σ(L¯))𝐹𝜎¯𝐿\textstyle{F(\sigma(\overline{L}))}

By Proposition 3.11, 𝒮pan2×(𝒞)𝒮superscriptsubscriptpan2𝒞\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right) is semistrict for any \infty-category 𝒞𝒞\mathcal{C} having finite limits, in particular when 𝒞=N(C)𝒞𝑁𝐶\mathcal{C}=N(C) as we shall assume for the remainder of this section333In fact, we only ever make use of the particular case of C=(FinΔ)op𝐶superscriptsubscriptFinΔopC=({\rm Fin}_{\Delta})^{\rm op}. Performing a similar analysis as we present in this section for a general \infty-category would involve explicitly determining certain colimits in 𝒞at𝒞subscriptat\mathcal{C}{\rm at}_{\infty}. This is both considerably more difficult and unnecessary for our purposes.. We can therefore apply the above to compute its unstraightening.

Remark 3.13.

As a slight abuse of notation we will not distinguish between an element (f,ϕ)Nk(Fin×Δop)𝑓italic-ϕsubscript𝑁𝑘subscriptFinsuperscriptΔop(f,\phi)\in N_{k}({\rm Fin}_{*}\times\Delta^{\rm op}) and its opposite functor

(f,ϕ):[k]opFinop×Δ.:𝑓italic-ϕsuperscriptdelimited-[]𝑘opsuperscriptsubscriptFinopΔ(f,\phi):[k]^{\rm op}\to{\rm Fin}_{*}^{\rm op}\times\Delta.

Furthermore, for jj[k]𝑗superscript𝑗delimited-[]𝑘j\leq j^{\prime}\in[k] we write the composite morphisms as

fj,j:f(j)f(j)Finandϕj,j:ϕ(j)ϕ(j)Δ.:subscript𝑓superscript𝑗𝑗𝑓𝑗𝑓superscript𝑗subscriptFinandsubscriptitalic-ϕsuperscript𝑗𝑗:italic-ϕsuperscript𝑗italic-ϕ𝑗Δf_{j^{\prime},j}:f(j)\to f(j^{\prime})\in{\rm Fin}_{*}\ \mathrm{and}\ \phi_{j^{\prime},j}:\phi(j^{\prime})\to\phi(j)\in\Delta.

Let ΔinjsubscriptΔinj\Delta_{\mathrm{inj}} be the wide subcategory of ΔΔ\Delta on the injective maps and let I(k)𝐼𝑘I(k) denote the full subcategory of [k]op×Δinj/[k]superscriptdelimited-[]𝑘opsubscriptΔinjdelimited-[]𝑘[k]^{\rm op}\times\Delta_{\mathrm{inj}/{[k]}} on those objects (j,J)𝑗𝐽(j,J) such that J[j]𝐽delimited-[]𝑗\emptyset\neq J\subset[j]. For each (f,ϕ)Nk(Fin×Δop)𝑓italic-ϕsubscript𝑁𝑘subscriptFinsuperscriptΔop(f,\phi)\in N_{k}({\rm Fin}_{*}\times\Delta^{\rm op}), denote by E(f,ϕ)𝐸𝑓italic-ϕE(f,\phi) the composite

I(k)𝐼𝑘\textstyle{I(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}[k]op×Δinj/[k]superscriptdelimited-[]𝑘opsubscriptΔinjdelimited-[]𝑘\textstyle{[k]^{\rm op}\times\Delta_{\mathrm{inj}/{[k]}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}(f,ϕ)×O𝑓italic-ϕ𝑂\scriptstyle{(f,\phi)\times O}Finop×Δ2superscriptsubscriptFinopsuperscriptΔ2\textstyle{{\rm Fin}_{*}^{\rm op}\times\Delta^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Π()op×Σ,ΠsuperscriptopsuperscriptΣ\scriptstyle{\Pi({-})^{\rm op}\times\Sigma^{\bullet,\bullet}}Cat,Cat\textstyle{{\rm Cat},}

where O𝑂O is the forgetful map from Δinj/[k]subscriptΔinjdelimited-[]𝑘\Delta_{\mathrm{inj}/{[k]}} to ΔΔ\Delta.

Lemma 3.14.

The k𝑘k-simplices of the unstraightening of 𝒮pan2×(N(C))𝒮superscriptsubscriptpan2𝑁𝐶\mathcal{S}{\rm pan}_{2}^{\times}\left({N(C)}\right) are the pairs

((f,ϕ)Nk(Fin×Δop),τ:E(f,ϕ)constC),:𝑓italic-ϕsubscript𝑁𝑘subscriptFinsuperscriptΔop𝜏𝐸𝑓italic-ϕsubscriptconst𝐶\left((f,\phi)\in N_{k}({\rm Fin}_{*}\times\Delta^{\rm op}),\ \tau:E(f,\phi)\Rightarrow{\rm const}_{C}\right),

such that each component τj,J:Π(f(j))op×Σϕ(j),JC:subscript𝜏𝑗𝐽Πsuperscript𝑓𝑗opsuperscriptΣitalic-ϕ𝑗𝐽𝐶\tau_{j,J}:\Pi({f(j)})^{\rm op}\times\Sigma^{\phi(j),J}\to C is cartesian and vertically constant.

Proof.

By Eq. 3.4 and Proposition 3.11, a k𝑘k-simplex in the unstraightening of 𝒮pan2×(N(C))𝒮superscriptsubscriptpan2𝑁𝐶\mathcal{S}{\rm pan}_{2}^{\times}\left({N(C)}\right) is an element (f,ϕ)Nk(Fin×Δop)𝑓italic-ϕsubscript𝑁𝑘subscriptFinsuperscriptΔop(f,\phi)\in N_{k}({\rm Fin}_{*}\times\Delta^{\rm op}) along with, for each JΔinj/[k]𝐽subscriptΔinjdelimited-[]𝑘J\in\Delta_{\mathrm{inj}/{[k]}}, a cartesian and vertically constant functor

τ(J):Π(f(J¯))op×Σϕ(J¯),JC:𝜏𝐽Πsuperscript𝑓¯𝐽opsuperscriptΣitalic-ϕ¯𝐽𝐽𝐶\tau(J):\Pi({f(\overline{J})})^{\rm op}\times\Sigma^{\phi(\overline{J}),J}\to C

such that for JLΔinj/[k]𝐽𝐿subscriptΔinjdelimited-[]𝑘J\to L\in\Delta_{\mathrm{inj}/{[k]}} the following diagram commutes

Π(f(L¯))op×Σϕ(L¯),JΠsuperscript𝑓¯𝐿opsuperscriptΣitalic-ϕ¯𝐿𝐽\textstyle{\Pi({f(\overline{L})})^{\rm op}\times\Sigma^{\phi(\overline{L}),J}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Π(f(J¯))op×Σϕ(J¯),JΠsuperscript𝑓¯𝐽opsuperscriptΣitalic-ϕ¯𝐽𝐽\textstyle{\Pi({f(\overline{J})})^{\rm op}\times\Sigma^{\phi(\overline{J}),J}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}τ(J)𝜏𝐽\scriptstyle{\tau(J)}Π(f(L¯))op×Σϕ(L¯),LΠsuperscript𝑓¯𝐿opsuperscriptΣitalic-ϕ¯𝐿𝐿\textstyle{\Pi({f(\overline{L})})^{\rm op}\times\Sigma^{\phi(\overline{L}),L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}τ(L)𝜏𝐿\scriptstyle{\tau(L)}C𝐶\textstyle{C} (3.5)

where the functor along the top is induced by J¯L¯[k]¯𝐽¯𝐿delimited-[]𝑘\overline{J}\leq\overline{L}\in[k] and the functor along the left is induced by JL𝐽𝐿J\subset L. Note that for every string of n𝑛n composable morphisms in Δinj/[k]subscriptΔinjdelimited-[]𝑘\Delta_{\mathrm{inj}/{[k]}} one can build an analogous commutative (n+1)𝑛1(n+1)-cube based on the commutativity of diagram 3.5 and the functoriality of Π()op×Σ,ΠsuperscriptopsuperscriptΣ\Pi({-})^{\rm op}\times\Sigma^{\bullet,\bullet}.

From this family {τ(J)}JΔinj/[k]subscript𝜏𝐽𝐽subscriptΔinjdelimited-[]𝑘\{\tau(J)\}_{J\in\Delta_{\mathrm{inj}/{[k]}}} we shall construct a natural transformation τ:E(f,ϕ)constC:𝜏𝐸𝑓italic-ϕsubscriptconst𝐶\tau:E(f,\phi)\Rightarrow{\rm const}_{C}. For each (j,J)I(k)𝑗𝐽𝐼𝑘(j,J)\in I(k) one has a morphism J[j]Δinj/[k]𝐽delimited-[]𝑗subscriptΔinjdelimited-[]𝑘J\to[j]\in\Delta_{\mathrm{inj}/{[k]}}, and hence the diagram 3.5 with L𝐿L replaced by [j]delimited-[]𝑗[j] commutes. We define τj,Jsubscript𝜏𝑗𝐽\tau_{j,J} to be the composite functor along the diagonal of this diagram, which is cartesian and vertically constant by construction. Given a morphism (j,J)(l,L)I(k)𝑗𝐽𝑙𝐿𝐼𝑘(j,J)\to(l,L)\in I(k) there is a triple of composable morphisms

JL[l][j]Δinj/[k].𝐽𝐿delimited-[]𝑙delimited-[]𝑗subscriptΔinjdelimited-[]𝑘J\to L\to[l]\to[j]\in\Delta_{\mathrm{inj}/{[k]}}.

The naturality of τ𝜏\tau is a consequence of the commutative 444-cube constructed from this triple of composable morphisms.

Conversely, given a natural transformation τ:E(f,ϕ)constC:𝜏𝐸𝑓italic-ϕsubscriptconst𝐶\tau:E(f,\phi)\Rightarrow{\rm const}_{C} having cartesian and vertically constant components, define τ(J):=τJ¯,Jassign𝜏𝐽subscript𝜏¯𝐽𝐽\tau(J):=\tau_{\overline{J},J}. The commutativity of diagram 3.5 follows directly from the naturality of τ𝜏\tau. ∎

By the universal property of colimits in CatCat{\rm Cat}, a natural transformation E(f,ϕ)constC𝐸𝑓italic-ϕsubscriptconst𝐶E(f,\phi)\Rightarrow{\rm const}_{C} is equivalently a functor colimE(f,ϕ)Ccolim𝐸𝑓italic-ϕ𝐶\operatornamewithlimits{colim}E(f,\phi)\to C. Observe that the diagonal inclusion of I(k)𝐼𝑘I(k) into the full subcategory of ([k]op)2×Δinj/[k]superscriptsuperscriptdelimited-[]𝑘op2subscriptΔinjdelimited-[]𝑘([k]^{\rm op})^{2}\times\Delta_{\mathrm{inj}/{[k]}} on objects (i,j,J)𝑖𝑗𝐽(i,j,J), where J[j]𝐽delimited-[]𝑗J\subset[j], is final. Therefore colimE(f,ϕ)colim𝐸𝑓italic-ϕ\operatornamewithlimits{colim}E(f,\phi) is isomorphic to the product of the colimits over Π(f)op:[k]opCat:Πsuperscript𝑓opsuperscriptdelimited-[]𝑘opCat\Pi({f})^{\rm op}:[k]^{\rm op}\to{\rm Cat} and the diagram

I(k)[k]op×Δinj/[k]ϕ×OΔ2Σ,Cat.𝐼𝑘superscriptdelimited-[]𝑘opsubscriptΔinjdelimited-[]𝑘italic-ϕ𝑂superscriptΔ2superscriptΣCat\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 12.2396pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&&\crcr}}}\ignorespaces{\hbox{\kern-12.2396pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{I(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 12.2396pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 25.43967pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 25.43967pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{[k]^{\rm op}\times\Delta_{\mathrm{inj}/{[k]}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 87.76657pt\raise 6.1111pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{\phi\times O}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 114.54123pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 95.34116pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 114.54123pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\Delta^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 141.03578pt\raise 5.8361pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.8361pt\hbox{$\scriptstyle{\Sigma^{\bullet,\bullet}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 164.07474pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 144.87466pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 164.07474pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\rm Cat}$}}}}}}}\ignorespaces}}}}\ignorespaces.

It follows that

colimE(f,ϕ)Π(f(0))op×(colim(j,J)I(k)ΣL(ϕ)j,J),similar-to-or-equalscolim𝐸𝑓italic-ϕΠsuperscript𝑓0opsubscriptcolim𝑗𝐽𝐼𝑘superscriptΣ𝐿subscriptitalic-ϕ𝑗𝐽\operatornamewithlimits{colim}E(f,\phi)\simeq\Pi({f(0)})^{\rm op}\times\left(\operatornamewithlimits{colim}_{(j,J)\in I(k)}\Sigma^{L(\phi)_{j,J}}\right),

where L(ϕ)𝐿italic-ϕL(\phi) is the composite

L(ϕ):I(k)[k]op×Δinj/[k]ϕ×OΔ2×Cat.:𝐿italic-ϕ𝐼𝑘superscriptdelimited-[]𝑘opsubscriptΔinjdelimited-[]𝑘italic-ϕ𝑂superscriptΔ2CatL(\phi):\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 12.2396pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&&\crcr}}}\ignorespaces{\hbox{\kern-12.2396pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{I(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 12.2396pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 25.43967pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 25.43967pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{[k]^{\rm op}\times\Delta_{\mathrm{inj}/{[k]}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 87.76657pt\raise 6.1111pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{\phi\times O}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 114.54123pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 95.34116pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 114.54123pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\Delta^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 139.89688pt\raise 5.33333pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{\bullet\times\bullet}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 164.07474pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 144.87466pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 164.07474pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\rm Cat}$}}}}}}}\ignorespaces}}}}\ignorespaces. (3.6)

It follows from Corollary 3.4 that functors colimE(f,ϕ)Ccolim𝐸𝑓italic-ϕ𝐶\operatornamewithlimits{colim}E(f,\phi)\to C are equivalently functors Π(f(0))op×ΣcolimL(ϕ)CΠsuperscript𝑓0opsuperscriptΣcolim𝐿italic-ϕ𝐶\Pi({f(0)})^{\rm op}\times\Sigma^{\operatornamewithlimits{colim}L(\phi)}\to C.

Our first task in this section is to compute colimL(ϕ)colim𝐿italic-ϕ\operatornamewithlimits{colim}L(\phi). We shall then determine conditions under which a functor τ:Π(f(0))op×ΣcolimL(ϕ)C:𝜏Πsuperscript𝑓0opsuperscriptΣcolim𝐿italic-ϕ𝐶\tau:\Pi({f(0)})^{\rm op}\times\Sigma^{\operatornamewithlimits{colim}L(\phi)}\to C restricts to cartesian and vertically constant functors τj,Jsubscript𝜏𝑗𝐽\tau_{j,J}.

Computing the colimit of the diagram L(ϕ)𝐿italic-ϕL(\phi).

Recall that the Grothendieck construction of a functor G:DCat:𝐺𝐷CatG:D\to{\rm Cat} is the category having as objects

{(g,d)|dD,gG(d)},conditional-set𝑔𝑑formulae-sequence𝑑𝐷𝑔𝐺𝑑\left\{(g,d)\ |\ d\in D,\ g\in G(d)\right\},

and morphisms

(α,δ):(g,d)(g,d),δ:ddandα:G(δ)(d)d.:𝛼𝛿𝑔𝑑superscript𝑔superscript𝑑𝛿:𝑑superscript𝑑and𝛼:𝐺𝛿𝑑superscript𝑑(\alpha,\delta):(g,d)\to(g^{\prime},d^{\prime}),\ \delta:d\to d^{\prime}\ {\rm and}\ \alpha:G(\delta)(d)\to d^{\prime}.

Let Mϕsubscript𝑀italic-ϕM_{\phi}, for ϕNk(Δop)italic-ϕsubscript𝑁𝑘superscriptΔop\phi\in N_{k}(\Delta^{\rm op}), be the category which is the Grothendieck construction of the functor

[k]opϕΔCat.superscriptdelimited-[]𝑘opitalic-ϕΔCat\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 11.49376pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\crcr}}}\ignorespaces{\hbox{\kern-11.49376pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{[k]^{\rm op}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 18.40834pt\raise 6.1111pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{\phi}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 35.49376pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 35.49376pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\Delta\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 49.82712pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 73.8271pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 73.8271pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\rm Cat}$}}}}}}}\ignorespaces}}}}\ignorespaces.

Explicitly, Mϕsubscript𝑀italic-ϕM_{\phi} is the poset having object set

{(a,b)|b[k]op,aϕ(b)},conditional-set𝑎𝑏formulae-sequence𝑏superscriptdelimited-[]𝑘op𝑎italic-ϕ𝑏\{(a,b)\ |\ b\in[k]^{\rm op},a\in\phi(b)\},

and ordering defined by declaring (a,b)(a,b)𝑎𝑏superscript𝑎superscript𝑏(a,b)\leq(a^{\prime},b^{\prime}) if and only if bb[k]op𝑏superscript𝑏superscriptdelimited-[]𝑘opb\leq b^{\prime}\in[k]^{\rm op} and ϕb,b(a)asubscriptitalic-ϕ𝑏superscript𝑏𝑎superscript𝑎\phi_{b,b^{\prime}}(a)\leq a^{\prime}.

Example 3.15.

For ϕNk(Δop)italic-ϕsubscript𝑁𝑘superscriptΔop\phi\in N_{k}(\Delta^{\rm op}) the constant map on [n]delimited-[]𝑛[n], the poset Mϕsubscript𝑀italic-ϕM_{\phi} is [n]×[k]opdelimited-[]𝑛superscriptdelimited-[]𝑘op[n]\times[k]^{\rm op}.

Example 3.16.

For the unique active morphism ϕ=([2]|[1])N1(Δop)\phi=([2]\Mapstochar\leftarrow[1])\in N_{1}(\Delta^{\rm op}), the poset Mϕsubscript𝑀italic-ϕM_{\phi} is

(0,1)01\textstyle{(0,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}(1,1)11\textstyle{(1,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}(0,0)00\textstyle{(0,0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}(1,0)10\textstyle{(1,0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}(2,0)20\textstyle{(2,0)}
Lemma 3.17.

The poset Mϕsubscript𝑀italic-ϕM_{\phi} is the colimit of the diagram L(ϕ)𝐿italic-ϕL(\phi).

Proof.

As the category Mϕsubscript𝑀italic-ϕM_{\phi} is obtained via the Grothendieck construction it is the colimit of the diagram

Σ([k]op)[k]op×[k]ϕ×[k]/opCat.superscriptΣsuperscriptdelimited-[]𝑘opsuperscriptdelimited-[]𝑘opdelimited-[]𝑘italic-ϕsubscriptsuperscriptdelimited-[]𝑘opabsentCat\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 13.57918pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\crcr}}}\ignorespaces{\hbox{\kern-13.57918pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\Sigma^{([k]^{\rm op})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 26.77925pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 26.77925pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{[k]^{\rm op}\times[k]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 73.24774pt\raise 7.30556pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.80556pt\hbox{$\scriptstyle{\phi\times[k]^{\rm op}_{-/}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 103.2433pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 84.04323pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 103.2433pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\rm Cat}$}}}}}}}\ignorespaces}}}}\ignorespaces.

Observe that, since [k]i/op=[i]op[i]subscriptsuperscriptdelimited-[]𝑘op𝑖superscriptdelimited-[]𝑖opsimilar-to-or-equalsdelimited-[]𝑖[k]^{\rm op}_{i/}=[i]^{\rm op}\simeq[i], one can obtain this diagram by precomposing the diagram in Eq. 3.6 with the functor

F:Σ([k]op)I(k),[j;i](j,[i]).:𝐹formulae-sequencesuperscriptΣsuperscriptdelimited-[]𝑘op𝐼𝑘maps-to𝑗𝑖𝑗delimited-[]𝑖F:\Sigma^{([k]^{\rm op})}\to I(k),\quad[j;i]\mapsto(j,[i]).

The functor F𝐹F is final, as for any (j,J)I(k)𝑗𝐽𝐼𝑘(j,J)\in I(k) with J¯=maxJ¯𝐽max𝐽\overline{J}={\rm max}\,J, the category F(j,J)/superscript𝐹𝑗𝐽F^{(j,J)/} is the full subcategory of Σ([k]op)superscriptΣsuperscriptdelimited-[]𝑘op\Sigma^{([k]^{\rm op})} on those objects

{[a;b]|J¯baj},conditional-set𝑎𝑏¯𝐽𝑏𝑎𝑗\{[a;b]\ |\ \overline{J}\leq b\leq a\leq j\},

which is non-empty and connected. ∎

The cartesian and vertical constancy conditions.

Having determined that colimL(ϕ)=Mϕcolim𝐿italic-ϕsubscript𝑀italic-ϕ\operatornamewithlimits{colim}L(\phi)=M_{\phi} we shall now define conditions under which a functor τ:Π(f(0))op×ΣMϕC:𝜏Πsuperscript𝑓0opsuperscriptΣsubscript𝑀italic-ϕ𝐶\tau:\Pi({f(0)})^{\rm op}\times\Sigma^{M_{\phi}}\to C induces cartesian and vertically constant functors τj,J:Π(f(j))op×Σϕ(j),JC:subscript𝜏𝑗𝐽Πsuperscript𝑓𝑗opsuperscriptΣitalic-ϕ𝑗𝐽𝐶\tau_{j,J}:\Pi({f(j)})^{\rm op}\times\Sigma^{\phi(j),J}\to C for each (j,J)I(k)𝑗𝐽𝐼𝑘(j,J)\in I(k).

Definition 3.18.

Define Vϕsubscript𝑉italic-ϕV_{\phi} to be the sub-poset of Mϕsubscript𝑀italic-ϕM_{\phi} on those morphisms (a,b)(ϕb,b(a),b)𝑎𝑏subscriptitalic-ϕ𝑏superscript𝑏𝑎superscript𝑏(a,b)\to(\phi_{b,b^{\prime}}(a),b^{\prime}) and ΛMϕsuperscriptΛsubscript𝑀italic-ϕ\Lambda^{M_{\phi}} to be the full sub-poset of ΣMϕsuperscriptΣsubscript𝑀italic-ϕ\Sigma^{M_{\phi}} on those intervals [(a,b);(ϕb,b(a),b)]𝑎𝑏subscriptitalic-ϕ𝑏superscript𝑏superscript𝑎superscript𝑏[(a,b);(\phi_{b,b^{\prime}}(a^{\prime}),b^{\prime})] satisfying |bb|1superscript𝑏𝑏1|b^{\prime}-b|\leq 1 and |aa|1superscript𝑎𝑎1|a^{\prime}-a|\leq 1.

Example 3.19.

For ϕitalic-ϕ\phi the constant map on [n]delimited-[]𝑛[n], one has that VϕVn,ksimilar-to-or-equalssubscript𝑉italic-ϕsubscript𝑉𝑛𝑘V_{\phi}\simeq V_{n,k} and ΛMϕΛn,ksimilar-to-or-equalssuperscriptΛsubscript𝑀italic-ϕsuperscriptΛ𝑛𝑘\Lambda^{M_{\phi}}\simeq\Lambda^{n,k}.

Example 3.20.

For the unique active morphism ϕ=([2]|[1])N1(Δop)\phi=([2]\Mapstochar\leftarrow[1])\in N_{1}(\Delta^{\rm op}), the poset Vϕsubscript𝑉italic-ϕV_{\phi} is

(0,1)01\textstyle{(0,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}(1,1)11\textstyle{(1,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}(0,0)00\textstyle{(0,0)}(1,0)10\textstyle{(1,0)}(2,0)20\textstyle{(2,0)}

and the poset ΛMϕsuperscriptΛsubscript𝑀italic-ϕ\Lambda^{M_{\phi}} is

[(0,1);(0,1)]0101\textstyle{[(0,1);(0,1)]}[(0,1);(1,1)]0111\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces[(0,1);(1,1)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}[(1,1);(1,1)]1111\textstyle{[(1,1);(1,1)]}[(0,1);(0,0)]0100\textstyle{[(0,1);(0,0)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}[(0,1);(2,0)]0120\textstyle{[(0,1);(2,0)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}[(1,1);(2,0)]1120\textstyle{[(1,1);(2,0)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}[(0,0);(0,0)]0000\textstyle{[(0,0);(0,0)]}[(0,0);(1,0)]0010\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces[(0,0);(1,0)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}[(1,0);(1,0)]1010\textstyle{[(1,0);(1,0)]}[(1,0);(2,0)]1020\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces[(1,0);(2,0)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}[(2,0);(2,0)]2020\textstyle{[(2,0);(2,0)]}
Definition 3.21.

Let C𝐶C be a category with finite limits and (f,ϕ)Nk(Fin×Δop)𝑓italic-ϕsubscript𝑁𝑘subscriptFinsuperscriptΔop(f,\phi)\in N_{k}({\rm Fin}_{*}\times\Delta^{\rm op}). Then we say a functor τ:Π(f(0))op×ΣMϕC:𝜏Πsuperscript𝑓0opsuperscriptΣsubscript𝑀italic-ϕ𝐶\tau:\Pi({f(0)})^{\rm op}\times\Sigma^{M_{\phi}}\to C is:

  1. 1.

    cartesian if it is the right Kan extension of its restriction to P(f(0))op×ΛMϕ𝑃superscript𝑓0opsuperscriptΛsubscript𝑀italic-ϕP({f(0)})^{\rm op}\times\Lambda^{M_{\phi}}.

  2. 2.

    vertically constant if morphisms of the form (Id,v)Id𝑣({\rm Id},v) for vΣVϕ𝑣superscriptΣsubscript𝑉italic-ϕv\in\Sigma^{V_{\phi}} are sent to isomorphisms.

Remark 3.22.

When (f,ϕ)Nk(Fin×Δop)𝑓italic-ϕsubscript𝑁𝑘subscriptFinsuperscriptΔop(f,\phi)\in N_{k}({\rm Fin}_{*}\times\Delta^{\rm op}) is the constant map on (S,[n])subscript𝑆delimited-[]𝑛(S_{*},[n]) this definition reduces to the notions already introduced for functors Π(S)op×Σn,kCΠsuperscript𝑆opsuperscriptΣ𝑛𝑘𝐶\Pi({S})^{\rm op}\times\Sigma^{n,k}\to C.

Since the categories Mϕsubscript𝑀italic-ϕM_{\phi} for various ϕNk(Δop)italic-ϕsubscript𝑁𝑘superscriptΔop\phi\in N_{k}(\Delta^{\rm op}) are obtained by the Grothendieck construction they are compatible in the following sense. For a natural transformation η:ϕϕ:𝜂superscriptitalic-ϕitalic-ϕ\eta:\phi^{\prime}\Rightarrow\phi one has a functor

M(η):MϕMϕ,(a,b)(ηb(a),b):𝑀𝜂formulae-sequencesubscript𝑀superscriptitalic-ϕsubscript𝑀italic-ϕmaps-to𝑎𝑏subscript𝜂𝑏𝑎𝑏M(\eta):M_{\phi^{\prime}}\to M_{\phi},\ (a,b)\mapsto(\eta_{b}(a),b)

and for a morphism γ:[n][k]:𝛾delimited-[]𝑛delimited-[]𝑘\gamma:[n]\to[k] there is a functor

M(γ):MϕγMϕ,(a,b)(a,γ(b)).:𝑀𝛾formulae-sequencesubscript𝑀italic-ϕ𝛾subscript𝑀italic-ϕmaps-to𝑎𝑏𝑎𝛾𝑏M(\gamma):M_{\phi\gamma}\to M_{\phi},\ (a,b)\mapsto(a,\gamma(b)).
Lemma 3.23.

Let C𝐶C be an category having finite limits, ϕNk(Δop)italic-ϕsubscript𝑁𝑘superscriptΔop\phi\in N_{k}(\Delta^{\rm op}) and F:ΣMϕC:𝐹superscriptΣsubscript𝑀italic-ϕ𝐶F:\Sigma^{M_{\phi}}\to C a cartesian functor. Then for a natural transformation η:ϕϕ:𝜂superscriptitalic-ϕitalic-ϕ\eta:\phi^{\prime}\Rightarrow\phi and a morphism γ:[n][k]:𝛾delimited-[]𝑛delimited-[]𝑘\gamma:[n]\to[k], the composite functors

ΣMϕsuperscriptΣsubscript𝑀superscriptitalic-ϕ\textstyle{\Sigma^{M_{\phi^{\prime}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}M(η)𝑀𝜂\scriptstyle{M(\eta)}ΣMϕsuperscriptΣsubscript𝑀italic-ϕ\textstyle{\Sigma^{M_{\phi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}F𝐹\scriptstyle{F}C𝐶\textstyle{C}andand\textstyle{{\rm and}}ΣMϕγsuperscriptΣsubscript𝑀italic-ϕ𝛾\textstyle{\Sigma^{M_{\phi\gamma}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}M(γ)𝑀𝛾\scriptstyle{M(\gamma)}ΣMϕsuperscriptΣsubscript𝑀italic-ϕ\textstyle{\Sigma^{M_{\phi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}F𝐹\scriptstyle{F}C𝐶\textstyle{C}

are cartesian.

The proof of this Lemma is somewhat technical and the details are unnecessary for the remainder of the text. We therefore separate its proof into Section 3.4.1.

Proposition 3.24.

Let C𝐶C be a category with finite limits, and let Cksubscriptdelimited-⟨⟩𝐶𝑘\left\langle{C}\right\rangle_{k} be the set

{((f,ϕ)Nk(Fin×Δop),τ:Π(f(0))op×ΣMϕC)|τcartesian,verticallyconstant}.\left\{\left((f,\phi)\in N_{k}({\rm Fin}_{*}\times\Delta^{\rm op}),\,\tau:\Pi({f(0)})^{\rm op}\times\Sigma^{M_{\phi}}\to C\right)\ |\ \tau\ {\rm cartesian},\ {\rm vertically}\ {\rm constant}\right\}.

Then the sets Cksubscriptdelimited-⟨⟩𝐶𝑘\left\langle{C}\right\rangle_{k} assemble into a sub simplicial set of the unstraightening of 𝒮pan2×(N(C))𝒮superscriptsubscriptpan2𝑁𝐶\mathcal{S}{\rm pan}_{2}^{\times}\left({N(C)}\right).

Remark 3.25.

In general, Cdelimited-⟨⟩𝐶\left\langle{C}\right\rangle is a proper sub simplicial set of the unstraightening. This will nonetheless suffice for our purposes as we will make use of this explicit form to map into the unstraightening.

Proof.

To begin, we observe that by Lemma 3.23, if τ𝜏\tau is cartesian then so is γτsuperscript𝛾𝜏\gamma^{*}\tau for any γ:[n][k]:𝛾delimited-[]𝑛delimited-[]𝑘\gamma:[n]\to[k], where

γτ:Π(fγ(0))op×ΣMϕγΠ(fγ(0),0)op×ΣM(γ)Π(f(0))op×ΣMϕτC.:superscript𝛾𝜏Πsuperscript𝑓𝛾0opsuperscriptΣsubscript𝑀italic-ϕ𝛾Πsuperscriptsubscript𝑓𝛾00opsuperscriptΣ𝑀𝛾Πsuperscript𝑓0opsuperscriptΣsubscript𝑀italic-ϕ𝜏𝐶\gamma^{*}\tau:\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 38.97261pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&\crcr}}}\ignorespaces{\hbox{\kern-38.97261pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\Pi({f\gamma(0)})^{\rm op}\times\Sigma^{M_{\phi\gamma}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 44.27806pt\raise 7.39166pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.89166pt\hbox{$\scriptstyle{\Pi({f_{\gamma(0),0}})^{\rm op}\times\Sigma^{M(\gamma)}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 104.97261pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 56.97261pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 80.97261pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 104.97261pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\Pi({f(0)})^{\rm op}\times\Sigma^{M_{\phi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 180.55376pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\tau}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 194.08379pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 194.08379pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{C}$}}}}}}}\ignorespaces}}}}\ignorespaces.

Furthermore, since M(γ)𝑀𝛾M(\gamma) maps Vϕγsubscript𝑉italic-ϕ𝛾V_{\phi\gamma} to Vϕsubscript𝑉italic-ϕV_{\phi}, the functor γτsuperscript𝛾𝜏\gamma^{*}\tau is vertically constant whenever τ𝜏\tau is. Therefore, the sets Cksubscriptdelimited-⟨⟩𝐶𝑘\left\langle{C}\right\rangle_{k} assemble into a simplicial set.

By Lemma 3.17 the set of k𝑘k-simplices of the unstraightening of 𝒮pan2×(N(C))𝒮superscriptsubscriptpan2𝑁𝐶\mathcal{S}{\rm pan}_{2}^{\times}\left({N(C)}\right) consists of pairs

{((f,ϕ)Nk(Fin×Δop),τ:Π(f(0))op×ΣMϕC)}\left\{\left((f,\phi)\in N_{k}({\rm Fin}_{*}\times\Delta^{\rm op}),\,\tau:\Pi({f(0)})^{\rm op}\times\Sigma^{M_{\phi}}\to C\right)\right\}

such that the induced functors τj,Jsubscript𝜏𝑗𝐽\tau_{j,J} are cartesian and vertically constant. To show that Cdelimited-⟨⟩𝐶\left\langle{C}\right\rangle is a sub simplicial set it therefore suffices to show that if τ𝜏\tau is cartesian and vertically constant than so are the functors τj,Jsubscript𝜏𝑗𝐽\tau_{j,J}. For the remainder of this proof we shall fix a cartesian, vertically constant functor τ𝜏\tau.

Observe that ϕ(j)×Jopitalic-ϕ𝑗superscript𝐽op\phi(j)\times J^{\rm op} is Mcϕ(j)subscript𝑀subscript𝑐italic-ϕ𝑗M_{c_{\phi(j)}}, where cϕ(j):JopCat:subscript𝑐italic-ϕ𝑗superscript𝐽opCatc_{\phi(j)}:J^{\rm op}\to{\rm Cat} is the constant functor on ϕ(j)italic-ϕ𝑗\phi(j), and that one has a diagram

Jopsuperscript𝐽op\textstyle{J^{\rm op}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}ρ𝜌\scriptstyle{\rho}cϕ(j)subscript𝑐italic-ϕ𝑗\scriptstyle{c_{\phi(j)}}η𝜂\scriptstyle{\eta}[k]opsuperscriptdelimited-[]𝑘op\textstyle{[k]^{\rm op}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}ϕitalic-ϕ\scriptstyle{\phi}Cat,Cat\textstyle{{\rm Cat}\,,}

where η𝜂\eta is the natural transformation having components ηi=ϕj,i:ϕ(j)ϕ(i):subscript𝜂𝑖subscriptitalic-ϕ𝑗𝑖italic-ϕ𝑗italic-ϕ𝑖\eta_{i}=\phi_{j,i}:\phi(j)\to\phi(i). Letting ψj,Jsubscript𝜓𝑗𝐽\psi_{j,J} denote the composite functor

ψj,J:ϕ(j)×JMcϕ(j)M(η)MϕρM(ρ)Mϕ,:subscript𝜓𝑗𝐽italic-ϕ𝑗𝐽similar-tosubscript𝑀subscript𝑐italic-ϕ𝑗𝑀𝜂subscript𝑀italic-ϕ𝜌𝑀𝜌subscript𝑀italic-ϕ\psi_{j,J}:\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 21.57785pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&&\crcr}}}\ignorespaces{\hbox{\kern-21.57785pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\phi(j)\times J\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 20.0556pt\raise 4.28406pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.28406pt\hbox{$\scriptstyle{\sim}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.97781pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 29.97781pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{M_{c_{\phi(j)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 55.2522pt\raise 6.5pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{M(\eta)}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 77.88931pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 63.48935pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 77.88931pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{M_{\phi\rho}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 101.00401pt\raise 6.5pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{M(\rho)}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 123.71283pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 109.31287pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 123.71283pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{M_{\phi}}$}}}}}}}\ignorespaces}}}}\ignorespaces,

one has that τj,Jsubscript𝜏𝑗𝐽\tau_{j,J} is

Π(f(j))op×Σϕ(j),JΠ(fj,0)op×Σψj,JΠ(f(0))op×ΣMϕτC.Πsuperscript𝑓𝑗opsuperscriptΣitalic-ϕ𝑗𝐽Πsuperscriptsubscript𝑓𝑗0opsuperscriptΣsubscript𝜓𝑗𝐽Πsuperscript𝑓0opsuperscriptΣsubscript𝑀italic-ϕ𝜏𝐶\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 39.18501pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\crcr}}}\ignorespaces{\hbox{\kern-39.18501pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\Pi({f(j)})^{\rm op}\times\Sigma^{\phi(j),J}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 48.55392pt\raise 7.13452pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.0377pt\hbox{$\scriptstyle{\Pi({f_{j,0}})^{\rm op}\times\Sigma^{\psi_{j,J}}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 105.18501pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 69.18501pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 105.18501pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\Pi({f(0)})^{\rm op}\times\Sigma^{M_{\phi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 186.76616pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\tau}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 206.29619pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 206.29619pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{C}$}}}}}}}\ignorespaces}}}}\ignorespaces.

It follows from Lemma 3.23 that τj,Jsubscript𝜏𝑗𝐽\tau_{j,J} is cartesian. The vertically constancy of τj,Jsubscript𝜏𝑗𝐽\tau_{j,J} follows from noting that M(η)𝑀𝜂M(\eta) sends Vcϕ(j)subscript𝑉subscript𝑐italic-ϕ𝑗V_{c_{\phi(j)}} to Vϕρsubscript𝑉italic-ϕ𝜌V_{\phi\rho} and M(ρ)𝑀𝜌M(\rho) sends Vϕρsubscript𝑉italic-ϕ𝜌V_{\phi\rho} to Vϕsubscript𝑉italic-ϕV_{\phi}. ∎

3.4.1 Proof of Lemma 3.23

Throughout this subsection we shall fix ϕ,ϕNk(Δop)italic-ϕsuperscriptitalic-ϕsubscript𝑁𝑘superscriptΔop\phi,\phi^{\prime}\in N_{k}(\Delta^{\rm op}), a natural transformation η:ϕϕ:𝜂superscriptitalic-ϕitalic-ϕ\eta:\phi^{\prime}\Rightarrow\phi, a morphism γ:[n][k]:𝛾delimited-[]𝑛delimited-[]𝑘\gamma:[n]\to[k] and a cartesian functor F:ΣMϕC:𝐹superscriptΣsubscript𝑀italic-ϕ𝐶F:\Sigma^{M_{\phi}}\to C. We shall also make use of some notational shorthands. Set α=ΛMϕ𝛼superscriptΛsubscript𝑀superscriptitalic-ϕ\alpha=\Lambda^{M_{\phi^{\prime}}}, A=ΣMϕ𝐴superscriptΣsubscript𝑀superscriptitalic-ϕA=\Sigma^{M_{\phi^{\prime}}}, θ=ΛMϕγ𝜃superscriptΛsubscript𝑀italic-ϕ𝛾\theta=\Lambda^{M_{\phi\gamma}}, Θ=ΣMϕγΘsuperscriptΣsubscript𝑀italic-ϕ𝛾\Theta=\Sigma^{M_{\phi\gamma}}, ω=ΛMϕ𝜔superscriptΛsubscript𝑀italic-ϕ\omega=\Lambda^{M_{\phi}}, Ω=ΣMϕΩsuperscriptΣsubscript𝑀italic-ϕ\Omega=\Sigma^{M_{\phi}}, ξ=ΣM(η)𝜉superscriptΣ𝑀𝜂\xi=\Sigma^{M(\eta)} and ζ=ΣM(γ)𝜁superscriptΣ𝑀𝛾\zeta=\Sigma^{M(\gamma)}.

We first show that Fξ𝐹𝜉F\circ\xi is cartesian. Since F𝐹F is cartesian, for each xA𝑥𝐴x\in A, the object Fξ(x)𝐹𝜉𝑥F\xi(x) is the limit of the diagram

ω(η)ξ(x)/ΩFC,𝜔superscript𝜂𝜉𝑥Ω𝐹𝐶\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 18.88687pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\crcr}}}\ignorespaces{\hbox{\kern-18.88687pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\omega(\eta)^{\xi(x)/}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 18.88689pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 42.88687pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 42.88687pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\Omega\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 62.3723pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.39166pt\hbox{$\scriptstyle{F}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 80.1091pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 80.1091pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{C}$}}}}}}}\ignorespaces}}}}\ignorespaces,

where ω(η)𝜔𝜂\omega(\eta) is the full subcategory of ΩΩ\Omega containing ω𝜔\omega as well as those objects p𝑝p such that pξ(q)𝑝𝜉𝑞p\geq\xi(q) for some qα𝑞𝛼q\in\alpha. On the other hand, the right Kan extension of the restriction of Fξ𝐹𝜉F\xi to α𝛼\alpha evaluated at xA𝑥𝐴x\in A is the limit of the diagram

αx/ξx/ω(η)ξ(x)/ΩFC.superscript𝛼𝑥superscript𝜉𝑥𝜔superscript𝜂𝜉𝑥Ω𝐹𝐶\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 9.19878pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\crcr}}}\ignorespaces{\hbox{\kern-9.19878pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\alpha^{x/}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 14.52448pt\raise 6.8611pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.5pt\hbox{$\scriptstyle{\xi^{x/}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 33.19878pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 33.19878pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\omega(\eta)^{\xi(x)/}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 70.97253pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 94.97252pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 94.97252pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\Omega\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 114.45795pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.39166pt\hbox{$\scriptstyle{F}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 132.19475pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 132.19475pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{C}$}}}}}}}\ignorespaces}}}}\ignorespaces.

To show that Fξ𝐹𝜉F\xi is cartesian it therefore suffices to show that for each xA𝑥𝐴x\in A, the functor ξx/superscript𝜉𝑥\xi^{x/} is initial ([21] IX.3). That is, we must show that for each yω(η)ξ(x)/𝑦𝜔superscript𝜂𝜉𝑥y\in\omega(\eta)^{\xi(x)/}, the poset

ξ/yx/={zα|zx,ξ(z)y}subscriptsuperscript𝜉𝑥absent𝑦conditional-set𝑧𝛼formulae-sequence𝑧𝑥𝜉𝑧𝑦\xi^{x/}_{/y}=\left\{z\in\alpha\ |\ z\geq x,\ \xi(z)\leq y\right\}

is non-empty and connected.

Letting y=[(c,d);(c,d)]ω(η)ξ(x)/𝑦𝑐𝑑superscript𝑐superscript𝑑𝜔superscript𝜂𝜉𝑥y=[(c,d);(c^{\prime},d^{\prime})]\in\omega(\eta)^{\xi(x)/} we define the following

c^=max{iϕ(d)|ηd(i)c}andc^=min{iϕ(d)|ηd(ϕd,d(i))c}.^𝑐𝑖conditionalsuperscriptitalic-ϕ𝑑subscript𝜂𝑑𝑖𝑐andsuperscript^𝑐𝑖conditionalsuperscriptitalic-ϕ𝑑subscript𝜂superscript𝑑subscriptsuperscriptitalic-ϕ𝑑superscript𝑑𝑖superscript𝑐\hat{c}=\max\{i\in\phi^{\prime}(d)\ |\ \eta_{d}(i)\leq c\}\ {\rm and}\ \hat{c}^{\prime}=\min\{i\in\phi^{\prime}(d)\ |\ \eta_{d^{\prime}}(\phi^{\prime}_{d,d^{\prime}}(i))\geq c^{\prime}\}.

There are a now two cases to consider. First, if c^c^^𝑐superscript^𝑐\hat{c}\leq\hat{c}^{\prime} then y^=[(c^,d);(ϕd,d(c^),d)]^𝑦^𝑐𝑑subscriptitalic-ϕ𝑑superscript𝑑superscript^𝑐superscript𝑑\hat{y}=[(\hat{c},d);(\phi_{d,d^{\prime}}(\hat{c}^{\prime}),d^{\prime})] is the maximal element of α𝛼\alpha satisfying ξ(y^)y𝜉^𝑦𝑦\xi(\hat{y})\leq y. Therefore ξ/yx/=α/y^x/subscriptsuperscript𝜉𝑥absent𝑦subscriptsuperscript𝛼𝑥absent^𝑦\xi^{x/}_{/y}=\alpha^{x/}_{/\hat{y}} and so ξ/yx/subscriptsuperscript𝜉𝑥absent𝑦\xi^{x/}_{/y} is non-empty and connected as it has terminal object y^^𝑦\hat{y}. Second, if c^c^superscript^𝑐^𝑐\hat{c}^{\prime}\leq\hat{c} then from the inequalities

ϕd,d(c)cϕd,d(ηd(c^))ϕd,d(ηd(c^))ϕd,d(c)subscriptitalic-ϕ𝑑superscript𝑑𝑐superscript𝑐subscriptitalic-ϕ𝑑superscript𝑑subscript𝜂𝑑superscript^𝑐subscriptitalic-ϕ𝑑superscript𝑑subscript𝜂𝑑^𝑐subscriptitalic-ϕ𝑑superscript𝑑𝑐\phi_{d,d^{\prime}}(c)\leq c^{\prime}\leq\phi_{d,d^{\prime}}\left(\eta_{d}(\hat{c}^{\prime})\right)\leq\phi_{d,d^{\prime}}\left(\eta_{d}(\hat{c})\right)\leq\phi_{d,d^{\prime}}(c)

one concludes that c=ϕd,d(c)superscript𝑐subscriptitalic-ϕ𝑑superscript𝑑𝑐c^{\prime}=\phi_{d,d^{\prime}}(c), ηd(c^)=ηd(c^)=csubscript𝜂𝑑superscript^𝑐subscript𝜂𝑑^𝑐𝑐\eta_{d}(\hat{c}^{\prime})=\eta_{d}(\hat{c})=c, and [c^;c^]superscript^𝑐^𝑐[\hat{c}^{\prime};\hat{c}] is the largest subinterval of ϕ(d)superscriptitalic-ϕ𝑑\phi^{\prime}(d) sent to [c;c]𝑐𝑐[c;c] under ηdsubscript𝜂𝑑\eta_{d}. The poset ξ/yx/subscriptsuperscript𝜉𝑥absent𝑦\xi^{x/}_{/y} is non-empty as it contains [(c^,d);(ϕd,d(c^),d][(\hat{c},d);(\phi^{\prime}_{d,d^{\prime}}(\hat{c}),d^{\prime}]. Letting [p;q]𝑝𝑞[p;q] denote the maximal subinterval of [c^;c^]superscript^𝑐^𝑐[\hat{c}^{\prime};\hat{c}] such that x[(p,d);(q,d)]𝑥𝑝𝑑𝑞𝑑x\leq[(p,d);(q,d)], one has that for any zξ/yx/𝑧subscriptsuperscript𝜉𝑥absent𝑦z\in\xi^{x/}_{/y} there is at least one r[p;q]𝑟𝑝𝑞r\in[p;q] such that z[(r,d);(r,d)]𝑧𝑟𝑑𝑟𝑑z\leq[(r,d);(r,d)]. As the poset ξ/yx/subscriptsuperscript𝜉𝑥absent𝑦\xi^{x/}_{/y} contains the connected sub-poset

{[(u,d);(v,d)]|[u;v][p;q],|uv|1}conditional-set𝑢𝑑𝑣𝑑formulae-sequence𝑢𝑣𝑝𝑞𝑢𝑣1\{[(u,d);(v,d)]\ |\ [u;v]\subset[p;q],\,|u-v|\leq 1\}

and each element maps into this sub-poset it follows that ξ/yx/subscriptsuperscript𝜉𝑥absent𝑦\xi^{x/}_{/y} is connected.

Next, we will show that Fζ𝐹𝜁F\circ\zeta is cartesian. The argument is quite similar to the above, and we shall recycle certain notation. It suffices to show that for each xΘ𝑥Θx\in\Theta and yω(γ)ζ(x)/𝑦𝜔superscript𝛾𝜁𝑥y\in\omega(\gamma)^{\zeta(x)/}, the poset

ζ/yx/={zθ|zx,ζ(z)y},subscriptsuperscript𝜁𝑥absent𝑦conditional-set𝑧𝜃formulae-sequence𝑧𝑥𝜁𝑧𝑦\zeta^{x/}_{/y}=\left\{z\in\theta\ |\ z\geq x,\ \zeta(z)\leq y\right\},

is non-empty and connected, where ω(γ)𝜔𝛾\omega(\gamma) is full subcategory of ΩΩ\Omega containing ω𝜔\omega and everything above the image of θ𝜃\theta under ζ𝜁\zeta. Set y=[(c,d);(c,d)]ω(γ)ξ(x)/𝑦𝑐𝑑superscript𝑐superscript𝑑𝜔superscript𝛾𝜉𝑥y=[(c,d);(c^{\prime},d^{\prime})]\in\omega(\gamma)^{\xi(x)/} and

d^=max{i[n]op|γ(i)d[k]op}andd^=min{i[n]op|γ(i)d[k]op}.^𝑑𝑖conditionalsuperscriptdelimited-[]𝑛op𝛾𝑖𝑑superscriptdelimited-[]𝑘opandsuperscript^𝑑𝑖conditionalsuperscriptdelimited-[]𝑛op𝛾𝑖superscript𝑑superscriptdelimited-[]𝑘op\hat{d}=\max\{i\in[n]^{\rm op}\ |\gamma(i)\leq d\in[k]^{\rm op}\}\ {\rm and}\ \hat{d}^{\prime}=\min\{i\in[n]^{\rm op}\ |\ \gamma(i)\geq d^{\prime}\in[k]^{\rm op}\}.

There are again two cases. The first is when d^d^^𝑑superscript^𝑑\hat{d}\leq\hat{d}^{\prime}, then y^=[(c,d^);(c,d^)]^𝑦𝑐^𝑑superscript𝑐superscript^𝑑\hat{y}=[(c,\hat{d});(c^{\prime},\hat{d}^{\prime})] is the maximal element of θ𝜃\theta satisfying ζ(y^)y𝜁^𝑦𝑦\zeta(\hat{y})\leq y, proving as above that ζ/yx/subscriptsuperscript𝜁𝑥absent𝑦\zeta^{x/}_{/y} is non-empty and connected. Next, if d^d^superscript^𝑑^𝑑\hat{d}^{\prime}\leq\hat{d} then d=dsuperscript𝑑𝑑d^{\prime}=d and [d^;d^]superscript^𝑑^𝑑[\hat{d}^{\prime};\hat{d}] is the largest subinterval of [n]opsuperscriptdelimited-[]𝑛op[n]^{\rm op} sent to [d;d]𝑑𝑑[d;d]. The same analysis as above, mutatis mutundi, shows that ζ/yx/subscriptsuperscript𝜁𝑥absent𝑦\zeta^{x/}_{/y} is non-empty and connected.

4 Simplicial objects define lax algebras in 𝒮pan2×(𝒞)𝒮superscriptsubscriptpan2𝒞\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right)

In this section we present the main results of this paper. We first, in Section 4.1, explicitly construct a symmetric monoidal lax functor

𝒜lgα𝒮pan2×((inΔ)op),𝒜superscriptlgcoproduct𝛼𝒮superscriptsubscriptpan2superscriptsubscriptindouble-struck-Δop\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 12.7389pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-12.7389pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathcal{A}{\rm lg}^{\amalg}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 19.49995pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\alpha}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 36.7389pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}{\hbox{\kern 36.7389pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathcal{S}{\rm pan}_{2}^{\times}\left({(\mathcal{F}{\rm in}_{\mathbb{\Delta}})^{\rm op}}\right)}$}}}}}}}\ignorespaces}}}}\ignorespaces,

endowing the standard 111-simplex Δ[1]Δdelimited-[]1\Delta\left[{1}\right] with the structure of a lax algebra. Then, in Section 4.2, we show how the object of 111-simplices 𝒳1subscript𝒳1\mathcal{X}_{1} of a simplicial object 𝒳𝒞Δ𝒳subscript𝒞double-struck-Δ\mathcal{X}\in\mathcal{C}_{\mathbb{\Delta}} in an \infty-category having finite limits inherits the same structure from the universal property of Δ[]Δdelimited-[]\Delta\left[{\bullet}\right]. Finally, we show in Section 4.3 that 𝒳𝒳\mathcal{X} satisfies the 222-Segal condition if and only if it inherits an algebra structure from Δ[]Δdelimited-[]\Delta\left[{\bullet}\right].

4.1 The lax algebra structure on Δ[1]Δdelimited-[]1\Delta\left[{1}\right]

Our first step in the construction of the lax algebra coming from a general simplicial object 𝒳𝒞Δsubscript𝒳subscript𝒞double-struck-Δ\mathcal{X}_{\bullet}\in\mathcal{C}_{\mathbb{\Delta}} will be to carry out the construction for the initial simplicial object Δ[](inΔ)opΔdelimited-[]superscriptsubscriptindouble-struck-Δop\Delta\left[{\bullet}\right]\in(\mathcal{F}{\rm in}_{\mathbb{\Delta}})^{\rm op}. According to Definitions 2.8 and 2.13, this amounts to constructing a morphism of fibrations

Un(𝒜lg)Un𝒜superscriptlgcoproduct\textstyle{{\rm Un}(\mathcal{A}{\rm lg}^{\amalg})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α𝛼\scriptstyle{\alpha}Un(𝒮pan2×((inΔ)op))Un𝒮superscriptsubscriptpan2superscriptsubscriptindouble-struck-Δop\textstyle{{\rm Un}\left(\mathcal{S}{\rm pan}_{2}^{\times}\left({(\mathcal{F}{\rm in}_{\mathbb{\Delta}})^{\rm op}}\right)\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}in×Δopsubscriptinsuperscriptdouble-struck-Δop\textstyle{\mathcal{F}{\rm in}_{*}\times\mathbb{\Delta}^{\rm op}}

preserving cocartesian lifts of morphisms of the form (f,ϕ)in×(Δin)op𝑓italic-ϕsubscriptinsuperscriptsubscriptdouble-struck-Δinop(f,\phi)\in\mathcal{F}{\rm in}_{*}\times(\mathbb{\Delta}_{\rm in})^{\rm op}. Furthermore, the image of the object 1¯¯1\underline{1} in the fibre over (1¯,[0])subscript¯1delimited-[]0(\underline{1}_{\ast},[0]) must be Δ[1]Δdelimited-[]1\Delta\left[{1}\right].

Before diving into the detailed construction of α𝛼\alpha, let us first give an informal description. On objects, the lax functor α𝛼\alpha is simply

α:XXΔ[1]=xXΔ[1].:𝛼maps-to𝑋𝑋Δdelimited-[]1subscriptcoproduct𝑥𝑋Δdelimited-[]1\alpha:X\mapsto X\cdot\Delta\left[{1}\right]=\coprod_{x\in X}\Delta\left[{1}\right].

Recall that a morphism in 𝒜lg𝒜superscriptlgcoproduct\mathcal{A}{\rm lg}^{\amalg} is a function p:XY:𝑝𝑋𝑌p:X\to Y along with a linear ordering of p1(y)superscript𝑝1𝑦p^{-1}(y) for each yY𝑦𝑌y\in Y. The lax functor α𝛼\alpha sends the morphism p𝑝p to the morphism α(p)𝒮pan2×((inΔ)op)𝛼𝑝𝒮superscriptsubscriptpan2superscriptsubscriptindouble-struck-Δop\alpha(p)\in\mathcal{S}{\rm pan}_{2}^{\times}\left({(\mathcal{F}{\rm in}_{\mathbb{\Delta}})^{\rm op}}\right) given by the diagram

yYΔ[|p1(y)|]subscriptcoproduct𝑦𝑌Δdelimited-[]superscript𝑝1𝑦\textstyle{\displaystyle\coprod_{y\in Y}\Delta\left[{|p^{-1}(y)|}\right]}XΔ[1]𝑋Δdelimited-[]1\textstyle{X\cdot\Delta\left[{1}\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}σ𝜎\scriptstyle{\sigma}YΔ[1],𝑌Δdelimited-[]1\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces Y\cdot\Delta\left[{1}\right]\,,}λ𝜆\scriptstyle{\lambda}

The morphism λ𝜆\lambda sends the 111-simplex associated to the element yY𝑦𝑌y\in Y to the long edge of the standard simplex Δ[|p1(y)|]Δdelimited-[]superscript𝑝1𝑦\Delta\left[{|p^{-1}(y)|}\right]. Note that one can label the edges along the spine of Δ[|p1(y)|]Δdelimited-[]superscript𝑝1𝑦\Delta\left[{|p^{-1}(y)|}\right] by the elements of p1(y)superscript𝑝1𝑦p^{-1}(y) using the linear ordering. The morphism σ𝜎\sigma sends the 111-simplex associated to xX𝑥𝑋x\in X to the appropriate edge along the spine of Δ[|p1(p(x))|]Δdelimited-[]superscript𝑝1𝑝𝑥\Delta\left[{|p^{-1}(p(x))|}\right].

Example 4.1.

For the morphism m:2¯1¯:𝑚¯2¯1m:\underline{2}\to\underline{1}, the morphism α(m)𝛼𝑚\alpha(m) in 𝒮pan2×((inΔ)op)𝒮superscriptsubscriptpan2superscriptsubscriptindouble-struck-Δop\mathcal{S}{\rm pan}_{2}^{\times}\left({(\mathcal{F}{\rm in}_{\mathbb{\Delta}})^{\rm op}}\right) is given by the diagram in Eq. 1.1.

Example 4.2.

Consider the morphism (mId):3¯=2¯1¯1¯1¯=2¯:coproduct𝑚Id¯3coproduct¯2¯1coproduct¯1¯1¯2(m\amalg{\rm Id}):\underline{3}=\underline{2}\amalg\underline{1}\to\underline{1}\amalg\underline{1}=\underline{2}. Then α(mId)𝛼coproduct𝑚Id\alpha(m\amalg{\rm Id}) is given by the diagram

[Uncaptioned image]

The lax structure on the functor α𝛼\alpha is given by associating to each pair of composable morphisms X0subscript𝑋0\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}p1subscript𝑝1\scriptstyle{p_{1}}X1subscript𝑋1\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}p2subscript𝑝2\scriptstyle{p_{2}}X2subscript𝑋2\textstyle{X_{2}} in 𝒜lg𝒜superscriptlgcoproduct\mathcal{A}{\rm lg}^{\amalg} a 222-morphism in 𝒮pan2×((inΔ)op)𝒮superscriptsubscriptpan2superscriptsubscriptindouble-struck-Δop\mathcal{S}{\rm pan}_{2}^{\times}\left({(\mathcal{F}{\rm in}_{\mathbb{\Delta}})^{\rm op}}\right) of the form

α(p2p1)𝛼subscript𝑝2subscript𝑝1\textstyle{\alpha(p_{2}p_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α(X0)𝛼subscript𝑋0\textstyle{\alpha(X_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α(p2p1)𝛼subscript𝑝2subscript𝑝1\textstyle{\alpha(p_{2}p_{1})}α(X2).𝛼subscript𝑋2\textstyle{\alpha(X_{2})\ .\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α(p2)α(X1)α(p1)𝛼subscript𝑝2subscriptcoproduct𝛼subscript𝑋1𝛼subscript𝑝1\textstyle{\alpha(p_{2})\coprod_{\alpha(X_{1})}\alpha(p_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}
Example 4.3.

Consider the pair of composable morphisms 3¯¯3\textstyle{\underline{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}mIdcoproduct𝑚Id\scriptstyle{m\amalg{\rm Id}}2¯¯2\textstyle{\underline{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}m𝑚\scriptstyle{m}1¯¯1\textstyle{\underline{1}}. The lax structure on α𝛼\alpha is given by the diagram

[Uncaptioned image]
Outline of the construction.

Recall from Section 2.3 that 𝒜lg𝒜superscriptlgcoproduct\mathcal{A}{\rm lg}^{\amalg} is semistrict by construction, and so it is straightforward to determine from Eq. 3.4 that its unstraightening is given by

Un(𝒜lg)k={((f,ϕ)Nk(Fin×Δop),θ:Π(f(0))×ϕ(0)Alg)|θcocartesian}.{\rm Un}(\mathcal{A}{\rm lg}^{\amalg})_{k}=\left\{\left((f,\phi)\in N_{k}({\rm Fin}_{*}\times\Delta^{\rm op}),\ \theta:\Pi({f(0)})\times\phi(0)\to{\rm Alg}\right)\ |\ \theta\ {\rm cocartesian}\right\}.

From such data we will build a cartesian and vertically constant functor

α(θ):Π(f(0))op×ΣMϕ(FinΔ)op.:𝛼𝜃Πsuperscript𝑓0opsuperscriptΣsubscript𝑀italic-ϕsuperscriptsubscriptFinΔop\alpha(\theta):\Pi({f(0)})^{\rm op}\times\Sigma^{M_{\phi}}\to({\rm Fin}_{\Delta})^{\rm op}.

By Proposition 3.24 this defines a k𝑘k-simplex in the unstraightening of 𝒮pan2×((inΔ)op)𝒮superscriptsubscriptpan2superscriptsubscriptindouble-struck-Δop\mathcal{S}{\rm pan}_{2}^{\times}\left({(\mathcal{F}{\rm in}_{\mathbb{\Delta}})^{\rm op}}\right). To better understand the approach we take to the construction of α(θ)𝛼𝜃\alpha(\theta) it will be instructive to describe a special case.

Consider ((f,ϕ),θ)Un(𝒜lg)1𝑓italic-ϕ𝜃Unsubscript𝒜superscriptlgcoproduct1\left((f,\phi),\theta\right)\in{\rm Un}(\mathcal{A}{\rm lg}^{\amalg})_{1} where f𝑓f is constant on 1¯subscript¯1\underline{1}_{\ast} and ϕ:[2]|[1]\phi:[2]\Mapstochar\leftarrow[1] is the unique active morphism. Then the functor θ𝜃\theta is a pair of composable morphisms

X0p1X1p2X2.subscript𝑋0subscript𝑝1subscript𝑋1subscript𝑝2subscript𝑋2\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 8.93471pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\crcr}}}\ignorespaces{\hbox{\kern-8.93471pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 9.7738pt\raise 5.1875pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8264pt\hbox{$\scriptstyle{p_{1}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 22.13478pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 22.13478pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 40.84329pt\raise 5.1875pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8264pt\hbox{$\scriptstyle{p_{2}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 53.20427pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 53.20427pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{X_{2}}$}}}}}}}\ignorespaces}}}}\ignorespaces.

Since α(θ)𝛼𝜃\alpha(\theta) is cartesian it is determined by its restriction to ΛMϕsuperscriptΛsubscript𝑀italic-ϕ\Lambda^{M_{\phi}}, the diagram described in Example 3.20. The restriction of α(θ)𝛼𝜃\alpha(\theta) to ΛMϕsuperscriptΛsubscript𝑀italic-ϕ\Lambda^{M_{\phi}} is

α(X0)𝛼subscript𝑋0\textstyle{\alpha(X_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α(p2p1)𝛼subscript𝑝2subscript𝑝1\textstyle{\alpha(p_{2}p_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α(X2)𝛼subscript𝑋2\textstyle{\alpha(X_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α(X0)𝛼subscript𝑋0\textstyle{\alpha(X_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α(p2p1)𝛼subscript𝑝2subscript𝑝1\textstyle{\alpha(p_{2}p_{1})}α(X2)𝛼subscript𝑋2\textstyle{\alpha(X_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α(X0)𝛼subscript𝑋0\textstyle{\alpha(X_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α(p1)𝛼subscript𝑝1\textstyle{\alpha(p_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α(X1)𝛼subscript𝑋1\textstyle{\alpha(X_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α(p2)𝛼subscript𝑝2\textstyle{\alpha(p_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α(X2)𝛼subscript𝑋2\textstyle{\alpha(X_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}

Observe that all of the data in this diagram can be obtained from the following diagram

α(p2p1)𝛼subscript𝑝2subscript𝑝1\textstyle{\alpha(p_{2}p_{1})}α(p1)𝛼subscript𝑝1\textstyle{\alpha(p_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α(p2)𝛼subscript𝑝2\textstyle{\alpha(p_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α(X0)𝛼subscript𝑋0\textstyle{\alpha(X_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α(X1)𝛼subscript𝑋1\textstyle{\alpha(X_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α(X2)𝛼subscript𝑋2\textstyle{\alpha(X_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces} (4.1)

which is a functor α¯(θ):Σϕ(0)(FinΔ)op:¯𝛼𝜃superscriptΣitalic-ϕ0superscriptsubscriptFinΔop\overline{\alpha}(\theta):\Sigma^{\phi(0)}\to({\rm Fin}_{\Delta})^{\rm op}. Specifically, the restriction of α(θ)𝛼𝜃\alpha(\theta) to ΛMϕsuperscriptΛsubscript𝑀italic-ϕ\Lambda^{M_{\phi}} is obtained from α¯(θ)¯𝛼𝜃\overline{\alpha}(\theta) by restricting along a functor ΛMϕΣϕ(0)superscriptΛsubscript𝑀italic-ϕsuperscriptΣitalic-ϕ0\Lambda^{M_{\phi}}\to\Sigma^{\phi(0)}.

In general, observe that for each ϕNk(Δop)italic-ϕsubscript𝑁𝑘superscriptΔop\phi\in N_{k}(\Delta^{\rm op}) one has a natural transformation ϕcϕ(0)italic-ϕsubscript𝑐italic-ϕ0\phi\Rightarrow c_{\phi(0)} having components ϕi,0subscriptitalic-ϕ𝑖0\phi_{i,0}. The naturality of the Grothendieck construction implies that this natural transformation induces a functor

pϕ:Mϕϕ(0).:subscript𝑝italic-ϕsubscript𝑀italic-ϕitalic-ϕ0p_{\phi}:M_{\phi}\to\phi(0).

Our construction of α(θ)𝛼𝜃\alpha(\theta) for general ((f,ϕ),θ)Un(𝒜lg)k𝑓italic-ϕ𝜃Unsubscript𝒜superscriptlgcoproduct𝑘((f,\phi),\theta)\in{\rm Un}(\mathcal{A}{\rm lg}^{\amalg})_{k} proceeds by first defining a functor

α¯(θ):P(f(0))op×Σϕ(0)(FinΔ)op:¯𝛼𝜃𝑃superscript𝑓0opsuperscriptΣitalic-ϕ0superscriptsubscriptFinΔop\overline{\alpha}(\theta):P({f(0)})^{\rm op}\times\Sigma^{\phi(0)}\to({\rm Fin}_{\Delta})^{\rm op}

generalising the one in Eq. 4.1. The functor α(θ)𝛼𝜃\alpha(\theta) is then defined so that the following diagram is a right Kan extension

Π(f(0))op×ΣMϕΠsuperscript𝑓0opsuperscriptΣsubscript𝑀italic-ϕ\textstyle{\Pi({f(0)})^{\rm op}\times\Sigma^{M_{\phi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α(θ)𝛼𝜃\scriptstyle{\alpha(\theta)}(FinΔ)opsuperscriptsubscriptFinΔop\textstyle{({\rm Fin}_{\Delta})^{\rm op}}P(f(0))op×ΛMϕ𝑃superscript𝑓0opsuperscriptΛsubscript𝑀italic-ϕ\textstyle{P({f(0)})^{\rm op}\times\Lambda^{M_{\phi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}P(f(0))op×ΣMϕ𝑃superscript𝑓0opsuperscriptΣsubscript𝑀italic-ϕ\textstyle{P({f(0)})^{\rm op}\times\Sigma^{M_{\phi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Id×ΣpϕIdsuperscriptΣsubscript𝑝italic-ϕ\scriptstyle{{\rm Id}\times\Sigma^{p_{\phi}}}P(f(0))op×Σϕ(0)𝑃superscript𝑓0opsuperscriptΣitalic-ϕ0\textstyle{P({f(0)})^{\rm op}\times\Sigma^{\phi(0)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α¯(θ)¯𝛼𝜃\scriptstyle{\overline{\alpha}(\theta)}(FinΔ)opsuperscriptsubscriptFinΔop\textstyle{({\rm Fin}_{\Delta})^{\rm op}\ignorespaces\ignorespaces\ignorespaces\ignorespaces} (4.2)

The functor α(θ)𝛼𝜃\alpha(\theta) is manifestly cartesian, and is vertically constant as every morphism in Vϕsubscript𝑉italic-ϕV_{\phi} is sent to the identity in ϕ(0)italic-ϕ0\phi(0) by pϕsubscript𝑝italic-ϕp_{\phi}.

Construction of the functor α¯(θ)¯𝛼𝜃\overline{\alpha}(\theta).

By Theorem 3.3 it suffices to define a normal oplax functor

α¯(θ):P(f(0))op×ϕ(0)sp((FinΔ)op).:¯𝛼𝜃𝑃superscript𝑓0opitalic-ϕ0spsuperscriptsubscriptFinΔop\overline{\alpha}(\theta):P({f(0)})^{\rm op}\times\phi(0)\nrightarrow{\rm sp}\left(({\rm Fin}_{\Delta})^{\rm op}\right).

Recall from Eq. 1.2 that \nabla is the category of spans of the form ndelimited-⟨⟩𝑛\textstyle{\langle n\rangle}kdelimited-⟨⟩𝑘\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\langle k\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}mdelimited-⟨⟩𝑚\textstyle{\langle m\rangle} in Δ+subscriptΔ\Delta_{+}. The category of levelwise finite nabla sets, denoted FinsubscriptFin{\rm Fin}_{\nabla} is the category of functors opFinsuperscriptopFin\nabla^{\rm op}\to{\rm Fin}. From a morphism p:XY:𝑝𝑋𝑌p:X\to Y in AlgAlg{\rm Alg} one can define the following diagram of levelwise finite nabla sets

yY[p1(y)]subscriptcoproduct𝑦𝑌superscript𝑝1𝑦\textstyle{\displaystyle\coprod_{y\in Y}\nabla\left[{p^{-1}(y)}\right]}X𝑋\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Y,𝑌\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces Y\,,}

where X𝑋X and Y𝑌Y are constant nabla sets and [n]𝑛\nabla\left[{n}\right] is the nabla set represented by ndelimited-⟨⟩𝑛\langle n\rangle. The first and second map arise, respectively, from the following morphisms in \nabla:

{x}𝑥\textstyle{\{x\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}p1(y)superscript𝑝1𝑦\textstyle{p^{-1}(y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}{x}𝑥\textstyle{\{x\}}p1(p(x))superscript𝑝1𝑝𝑥\textstyle{p^{-1}(p(x))}{y}𝑦\textstyle{\{y\}}p1(y).superscript𝑝1𝑦\textstyle{p^{-1}(y)\,.}

We can then apply the functor 𝔊superscript𝔊\mathfrak{G}^{*} of Eq. 1.3 to obtain a span in (FinΔ)opsuperscriptsubscriptFinΔop({\rm Fin}_{\Delta})^{\rm op}.

Now, define α¯(θ)¯𝛼𝜃\overline{\alpha}(\theta) on objects as

α¯(θ)(s,i)=𝔊θ(s,i)=θ(s,i)Δ[1].¯𝛼𝜃𝑠𝑖superscript𝔊𝜃𝑠𝑖𝜃𝑠𝑖Δdelimited-[]1\overline{\alpha}(\theta)(s,i)=\mathfrak{G}^{*}\theta(s,i)=\theta(s,i)\cdot\Delta\left[{1}\right].

To define α¯(θ)¯𝛼𝜃\overline{\alpha}(\theta) on a morphism f:(s,i)(s,j):𝑓𝑠𝑖𝑠𝑗f:(s,i)\to(s,j) in P(f(0))op×ϕ(0)𝑃superscript𝑓0opitalic-ϕ0P({f(0)})^{\rm op}\times\phi(0), one applies the above construction to the morphism θ(f)𝜃𝑓\theta(f) in AlgAlg{\rm Alg}, yielding a span in (FinΔ)opsuperscriptsubscriptFinΔop({\rm Fin}_{\Delta})^{\rm op}

α¯(θ)(f)¯𝛼𝜃𝑓\textstyle{\overline{\alpha}(\theta)(f)}α¯(θ)(s,i)¯𝛼𝜃𝑠𝑖\textstyle{\overline{\alpha}(\theta)(s,i)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α¯(θ)(s,j).¯𝛼𝜃𝑠𝑗\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\overline{\alpha}(\theta)(s,j)\,.}

From a pair of composable morphisms X𝑋\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}p𝑝\scriptstyle{p}Y𝑌\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}q𝑞\scriptstyle{q}Z𝑍\textstyle{Z} in AlgAlg{\rm Alg}, one has a commutative diagram of nabla sets

zZ[(qp)1(z)]subscriptcoproduct𝑧𝑍superscript𝑞𝑝1𝑧\textstyle{\displaystyle\coprod_{z\in Z}\nabla\left[{(qp)^{-1}(z)}\right]}X𝑋\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}yY[p1(y)]subscriptcoproduct𝑦𝑌superscript𝑝1𝑦\textstyle{\displaystyle\coprod_{y\in Y}\nabla\left[{p^{-1}(y)}\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}a𝑎\scriptstyle{a}Y𝑌\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}zZ[q1(z)]subscriptcoproduct𝑧𝑍superscript𝑞1𝑧\textstyle{\ \displaystyle\coprod_{z\in Z}\nabla\left[{q^{-1}(z)}\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}b𝑏\scriptstyle{b}Z,𝑍\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces Z\,,}

where a𝑎a and b𝑏b arise, respectively, from the following morphisms in \nabla:

p1(y)superscript𝑝1𝑦\textstyle{p^{-1}(y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}(qp)1(z)superscript𝑞𝑝1𝑧\textstyle{(qp)^{-1}(z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}p𝑝\scriptstyle{p}p1(y)superscript𝑝1𝑦\textstyle{p^{-1}(y)}(qp)1(q(y))superscript𝑞𝑝1𝑞𝑦\textstyle{(qp)^{-1}(q(y))}q1(z)superscript𝑞1𝑧\textstyle{q^{-1}(z)}(qp)1(z).superscript𝑞𝑝1𝑧\textstyle{(qp)^{-1}(z)\,.}

Applying the functor 𝔊superscript𝔊\mathfrak{G}^{*} of Eq. 1.3 yields a diagram in FinΔsubscriptFinΔ{\rm Fin}_{\Delta}.

For a pair of composable morphisms (s,i)𝑠𝑖\textstyle{(s,i)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}f𝑓\scriptstyle{f}(s,j)𝑠𝑗\textstyle{(s,j)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}g𝑔\scriptstyle{g}(s,k)𝑠𝑘\textstyle{(s,k)} in P(f(0))op×ϕ(0)𝑃superscript𝑓0opitalic-ϕ0P({f(0)})^{\rm op}\times\phi(0) we define the corresponding component A¯(θ)g,f¯𝐴subscript𝜃𝑔𝑓\overline{A}(\theta)_{g,f} of the oplax structure on α¯(θ)¯𝛼𝜃\overline{\alpha}(\theta) as follows. Applying the construction of the preceding paragraph to the image under θ𝜃\theta of the pair of composable morphisms one has, in particular, a commutative square

α¯(θ)(s,j)¯𝛼𝜃𝑠𝑗\textstyle{\overline{\alpha}(\theta)(s,j)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α¯(θ)(g)¯𝛼𝜃𝑔\textstyle{\overline{\alpha}(\theta)(g)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α¯(θ)(f)¯𝛼𝜃𝑓\textstyle{\overline{\alpha}(\theta)(f)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α¯(θ)(gf).¯𝛼𝜃𝑔𝑓\textstyle{\overline{\alpha}(\theta)(g\circ f)\,.} (4.3)

Then the component A¯(θ)g,f¯𝐴subscript𝜃𝑔𝑓\overline{A}(\theta)_{g,f} is the universal morphism

A¯(θ)g,f:α¯(θ)(g)α¯(θ)(s,j)α¯(θ)(f)α¯(θ)(gf),:¯𝐴subscript𝜃𝑔𝑓¯𝛼𝜃𝑔subscriptcoproduct¯𝛼𝜃𝑠𝑗¯𝛼𝜃𝑓¯𝛼𝜃𝑔𝑓\overline{A}(\theta)_{g,f}:\overline{\alpha}(\theta)(g)\coprod_{\overline{\alpha}(\theta)(s,j)}\overline{\alpha}(\theta)(f)\to\overline{\alpha}(\theta)(g\circ f),

induced by the diagram in Eq. 4.3.

Lemma 4.4.

For each ((f,ϕ),θ)Un(𝒜lg)k𝑓italic-ϕ𝜃Unsubscript𝒜superscriptlgcoproduct𝑘((f,\phi),\theta)\in{\rm Un}(\mathcal{A}{\rm lg}^{\amalg})_{k}, the above data defines a normal oplax functor and hence a functor

α¯(θ):P(f(0))op×Σϕ(0)(FinΔ)op.:¯𝛼𝜃𝑃superscript𝑓0opsuperscriptΣitalic-ϕ0superscriptsubscriptFinΔop\overline{\alpha}(\theta):P({f(0)})^{\rm op}\times\Sigma^{\phi(0)}\to({\rm Fin}_{\Delta})^{\rm op}.
Proof.

The unitality conditions A¯(θ)Id,f=Idα¯(θ)(f)=A¯(θ)f,Id¯𝐴subscript𝜃Id𝑓subscriptId¯𝛼𝜃𝑓¯𝐴subscript𝜃𝑓Id\overline{A}(\theta)_{{\rm Id},f}={\rm Id}_{\overline{\alpha}(\theta)(f)}=\overline{A}(\theta)_{f,{\rm Id}} are straightforward to verify. Given a triple w𝑤\textstyle{w\ignorespaces\ignorespaces\ignorespaces\ignorespaces}g𝑔\scriptstyle{g}x𝑥\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces}h\scriptstyle{h}y𝑦\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}i𝑖\scriptstyle{i}z𝑧\textstyle{z} of composable morphisms in P(f(0))op×ϕ(0)𝑃superscript𝑓0opitalic-ϕ0P({f(0)})^{\rm op}\times\phi(0), along similar lines as above one has a diagram in FinΔsubscriptFinΔ{\rm Fin}_{\Delta}

α¯(θ)(ihg)¯𝛼𝜃𝑖𝑔\textstyle{\overline{\alpha}(\theta)(ihg)}α¯(θ)(gh)¯𝛼𝜃𝑔\textstyle{\overline{\alpha}(\theta)(gh)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α¯(θ)(ih)¯𝛼𝜃𝑖\textstyle{\overline{\alpha}(\theta)(ih)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α¯(θ)(g)¯𝛼𝜃𝑔\textstyle{\overline{\alpha}(\theta)(g)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α¯(θ)(h)¯𝛼𝜃\textstyle{\overline{\alpha}(\theta)(h)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α¯(θ)(i)¯𝛼𝜃𝑖\textstyle{\overline{\alpha}(\theta)(i)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α¯(θ)(w)¯𝛼𝜃𝑤\textstyle{\overline{\alpha}(\theta)(w)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α¯(θ)(x)¯𝛼𝜃𝑥\textstyle{\overline{\alpha}(\theta)(x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α¯(θ)(y)¯𝛼𝜃𝑦\textstyle{\overline{\alpha}(\theta)(y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α¯(θ)(z)¯𝛼𝜃𝑧\textstyle{\overline{\alpha}(\theta)(z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}

Then the associativity condition,

A¯(θ)i,hg(Idα¯(θ)(i)α¯(θ)(y)A¯(θ)h,g)=A¯(θ)ih,g(A¯(θ)i,hα¯(θ)(x)Idα¯(θ)(g)),¯𝐴subscript𝜃𝑖𝑔subscriptId¯𝛼𝜃𝑖subscriptcoproduct¯𝛼𝜃𝑦¯𝐴subscript𝜃𝑔¯𝐴subscript𝜃𝑖𝑔¯𝐴subscript𝜃𝑖subscriptcoproduct¯𝛼𝜃𝑥subscriptId¯𝛼𝜃𝑔\overline{A}(\theta)_{i,hg}\circ\left({\rm Id}_{\overline{\alpha}(\theta)(i)}\coprod_{\overline{\alpha}(\theta)(y)}\overline{A}(\theta)_{h,g}\right)=\overline{A}(\theta)_{ih,g}\circ\left(\overline{A}(\theta)_{i,h}\coprod_{\overline{\alpha}(\theta)(x)}{\rm Id}_{\overline{\alpha}(\theta)(g)}\right),

holds since both sides of the equation are the universal morphism for the colimit of the bottom two rows of the above diagram. ∎

The morphism α𝛼\alpha is a symmetric monoidal lax functor.

The diagram in Eq. 4.2 defines, for each (f,ϕ)Nk(Fin×Δop)𝑓italic-ϕsubscript𝑁𝑘subscriptFinsuperscriptΔop(f,\phi)\in N_{k}({\rm Fin}_{*}\times\Delta^{\rm op}) a functor

R(f,ϕ):Funcart(P(f(0))op×Σϕ(0),(FinΔ)op)Funcart(Π(f(0))op×ΣMϕ,(FinΔ)op).:subscript𝑅𝑓italic-ϕsuperscriptFuncart𝑃superscript𝑓0opsuperscriptΣitalic-ϕ0superscriptsubscriptFinΔopsuperscriptFuncartΠsuperscript𝑓0opsuperscriptΣsubscript𝑀italic-ϕsuperscriptsubscriptFinΔopR_{(f,\phi)}:{\rm Fun}^{\rm cart}\left(P({f(0)})^{\rm op}\times\Sigma^{\phi(0)},({\rm Fin}_{\Delta})^{\rm op}\right)\to{\rm Fun}^{\rm cart}\left(\Pi({f(0)})^{\rm op}\times\Sigma^{M_{\phi}},({\rm Fin}_{\Delta})^{\rm op}\right).

We define α(θ)𝛼𝜃\alpha(\theta), for ((f,ϕ),θ)Un(𝒜lg)k𝑓italic-ϕ𝜃Unsubscript𝒜superscriptlgcoproduct𝑘((f,\phi),\theta)\in{\rm Un}(\mathcal{A}{\rm lg}^{\amalg})_{k}, to be

α(θ):=R(f,ϕ)α¯(θ)(FinΔ)opk,assign𝛼𝜃subscript𝑅𝑓italic-ϕ¯𝛼𝜃subscriptdelimited-⟨⟩superscriptsubscriptFinΔop𝑘\alpha(\theta):=R_{(f,\phi)}\overline{\alpha}(\theta)\in\left\langle{({\rm Fin}_{\Delta})^{\rm op}}\right\rangle_{k},

where (FinΔ)opdelimited-⟨⟩superscriptsubscriptFinΔop\left\langle{({\rm Fin}_{\Delta})^{\rm op}}\right\rangle is the sub simplicial set of the unstraightening of 𝒮pan2×((inΔ)op)𝒮superscriptsubscriptpan2superscriptsubscriptindouble-struck-Δop\mathcal{S}{\rm pan}_{2}^{\times}\left({(\mathcal{F}{\rm in}_{\mathbb{\Delta}})^{\rm op}}\right) from Proposition 3.24.

Lemma 4.5.

The assignment ((f,ϕ),θ)α(θ)maps-to𝑓italic-ϕ𝜃𝛼𝜃((f,\phi),\theta)\mapsto\alpha(\theta) defines a morphism of simplicial sets

α:Un(𝒜lg)(FinΔ)op.:𝛼Un𝒜superscriptlgcoproductdelimited-⟨⟩superscriptsubscriptFinΔop\alpha:{\rm Un}(\mathcal{A}{\rm lg}^{\amalg})\to\left\langle{({\rm Fin}_{\Delta})^{\rm op}}\right\rangle.
Proof.

Fix ((f,ϕ),θ)Un(𝒜lg)k𝑓italic-ϕ𝜃Unsubscript𝒜superscriptlgcoproduct𝑘((f,\phi),\theta)\in{\rm Un}(\mathcal{A}{\rm lg}^{\amalg})_{k} and a morphism γ:[n][k]:𝛾delimited-[]𝑛delimited-[]𝑘\gamma:[n]\to[k]. We must show that γα(θ)=α(γθ)superscript𝛾𝛼𝜃𝛼superscript𝛾𝜃\gamma^{*}\alpha(\theta)=\alpha(\gamma^{*}\theta), that is, that the following diagram commutes,

Π(fγ(0))op×ΣMϕγΠsuperscript𝑓𝛾0opsuperscriptΣsubscript𝑀italic-ϕ𝛾\textstyle{\Pi({f\gamma(0)})^{\rm op}\times\Sigma^{M_{\phi\gamma}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α(γθ)𝛼superscript𝛾𝜃\scriptstyle{\alpha(\gamma^{*}\theta)}Π(f(0))op×ΣMϕΠsuperscript𝑓0opsuperscriptΣsubscript𝑀italic-ϕ\textstyle{\Pi({f(0)})^{\rm op}\times\Sigma^{M_{\phi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α(θ)𝛼𝜃\scriptstyle{\alpha(\theta)}(FinΔ)op,superscriptsubscriptFinΔop\textstyle{({\rm Fin}_{\Delta})^{\rm op}\,,}

where γθsuperscript𝛾𝜃\gamma^{*}\theta is the composite

Π(fγ(0))×ϕγ(0)Π𝑓𝛾0italic-ϕ𝛾0\textstyle{\Pi({f\gamma(0)})\times\phi\gamma(0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Π(f(0))×ϕ(0)Π𝑓0italic-ϕ0\textstyle{\Pi({f(0)})\times\phi(0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}θ𝜃\scriptstyle{\theta}Alg.Alg\textstyle{{\rm Alg}\,.}

Letting P(f(0))op¯¯𝑃superscript𝑓0op\overline{P({f(0)})^{\rm op}} denote the full subcategory of Π(f(0))opΠsuperscript𝑓0op\Pi({f(0)})^{\rm op} containing P(f(0))op𝑃superscript𝑓0opP({f(0)})^{\rm op} as well as those Uf(0)𝑈𝑓0U\subset f(0) such that Ufγ(0),01(s)𝑈superscriptsubscript𝑓𝛾001𝑠U\subset f_{\gamma(0),0}^{-1}(s) for some sfγ(0)𝑠𝑓𝛾0s\in f\gamma(0), one has that the following diagram commutes

Funcart(Π(f(0))op×ΣMϕ,(FinΔ)op)superscriptFuncartΠsuperscript𝑓0opsuperscriptΣsubscript𝑀italic-ϕsuperscriptsubscriptFinΔop\textstyle{{\rm Fun}^{\rm cart}\left(\Pi({f(0)})^{\rm op}\times\Sigma^{M_{\phi}},({\rm Fin}_{\Delta})^{\rm op}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}γsuperscript𝛾\scriptstyle{\gamma^{*}}Funcart(Π(fγ(0))op×ΣMϕγ,(FinΔ)op)superscriptFuncartΠsuperscript𝑓𝛾0opsuperscriptΣsubscript𝑀italic-ϕ𝛾superscriptsubscriptFinΔop\textstyle{{\rm Fun}^{\rm cart}\left(\Pi({f\gamma(0)})^{\rm op}\times\Sigma^{M_{\phi\gamma}},({\rm Fin}_{\Delta})^{\rm op}\right)}Funcart(P(f(0))op¯×Σϕ(0),(FinΔ)op)superscriptFuncart¯𝑃superscript𝑓0opsuperscriptΣitalic-ϕ0superscriptsubscriptFinΔop\textstyle{{\rm Fun}^{\rm cart}\left(\overline{P({f(0)})^{\rm op}}\times\Sigma^{\phi(0)},({\rm Fin}_{\Delta})^{\rm op}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}R(f,ϕ)subscript𝑅𝑓italic-ϕ\scriptstyle{R_{(f,\phi)}}γsuperscript𝛾\scriptstyle{\gamma^{*}}Funcart(P(fγ(0))op×Σϕγ(0),(FinΔ)op)superscriptFuncart𝑃superscript𝑓𝛾0opsuperscriptΣitalic-ϕ𝛾0superscriptsubscriptFinΔop\textstyle{{\rm Fun}^{\rm cart}\left(P({f\gamma(0)})^{\rm op}\times\Sigma^{\phi\gamma(0)},({\rm Fin}_{\Delta})^{\rm op}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}R(fγ,ϕγ)subscript𝑅𝑓𝛾italic-ϕ𝛾\scriptstyle{R_{(f\gamma,\phi\gamma)}}

It therefore suffices to show that for each sfγ(0)𝑠𝑓𝛾0s\in f\gamma(0),

α¯(γθ)(s,)=tfγ(0),01(s)γα¯(θ)(t,)¯𝛼superscript𝛾𝜃𝑠subscriptcoproduct𝑡superscriptsubscript𝑓𝛾001𝑠superscript𝛾¯𝛼𝜃𝑡\overline{\alpha}(\gamma^{*}\theta)(s,-)=\coprod_{t\in f_{\gamma(0),0}^{-1}(s)}\gamma^{*}\overline{\alpha}(\theta)(t,-)

as normal oplax functors ϕγ(0)sp((FinΔ)op)italic-ϕ𝛾0spsuperscriptsubscriptFinΔop\phi\gamma(0)\nrightarrow{\rm sp}\left(({\rm Fin}_{\Delta})^{\rm op}\right). This follows from the fact that θ𝜃\theta is cocartesian. ∎

We have therefore constructed a morphism of fibrations

Un(𝒜lg)Un𝒜superscriptlgcoproduct\textstyle{{\rm Un}(\mathcal{A}{\rm lg}^{\amalg})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α𝛼\scriptstyle{\alpha}Un(𝒮pan2×((inΔ)op))Un𝒮superscriptsubscriptpan2superscriptsubscriptindouble-struck-Δop\textstyle{{\rm Un}\left(\mathcal{S}{\rm pan}_{2}^{\times}\left({(\mathcal{F}{\rm in}_{\mathbb{\Delta}})^{\rm op}}\right)\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}in×Δopsubscriptinsuperscriptdouble-struck-Δop\textstyle{\mathcal{F}{\rm in}_{*}\times\mathbb{\Delta}^{\rm op}}

Furthermore, the image of the object 1¯¯1\underline{1} in the fibre over (1¯,[0])subscript¯1delimited-[]0(\underline{1}_{*},[0]) is Δ[1]Δdelimited-[]1\Delta\left[{1}\right]. To show that the α𝛼\alpha endows Δ[1]Δdelimited-[]1\Delta\left[{1}\right] with a lax algebra structure it remains only to show that α𝛼\alpha defines a symmetric monoidal lax functor.

Proposition 4.6.

The morphism of simplicial sets α𝛼\alpha is a symmetric monoidal lax functor

α:𝒜lg𝒮pan2×((inΔ)op).:𝛼𝒜superscriptlgcoproduct𝒮superscriptsubscriptpan2superscriptsubscriptindouble-struck-Δop\alpha:\mathcal{A}{\rm lg}^{\amalg}\rightsquigarrow\mathcal{S}{\rm pan}_{2}^{\times}\left({(\mathcal{F}{\rm in}_{\mathbb{\Delta}})^{\rm op}}\right).

Hence, α𝛼\alpha endows the standard 111-simplex Δ[1]Δdelimited-[]1\Delta\left[{1}\right] with the structure of a lax algebra..

Proof.

We must show that for every ((f,ϕ),θ)Un(𝒜lg)1𝑓italic-ϕ𝜃Unsubscript𝒜superscriptlgcoproduct1\left((f,\phi),\theta\right)\in{\rm Un}(\mathcal{A}{\rm lg}^{\amalg})_{1} with ϕitalic-ϕ\phi inert the functor

α(θ)1,[1]:Π(f(1))op×Σϕ(1),[1](FinΔ)op:𝛼subscript𝜃1delimited-[]1Πsuperscript𝑓1opsuperscriptΣitalic-ϕ1delimited-[]1superscriptsubscriptFinΔop\alpha(\theta)_{1,[1]}:\Pi({f(1)})^{\rm op}\times\Sigma^{\phi(1),[1]}\to({\rm Fin}_{\Delta})^{\rm op}

defines an equivalence in the quasicategory 𝒮pf(1),ϕ(1)((inΔ)op)𝒮subscriptp𝑓1italic-ϕ1superscriptsubscriptindouble-struck-Δop\mathcal{S}{\rm p}_{f(1),\phi(1)}((\mathcal{F}{\rm in}_{\mathbb{\Delta}})^{\rm op}) ([19] 3.2.5.2).

For ϕitalic-ϕ\phi inert, the poset Mϕsubscript𝑀italic-ϕM_{\phi} is of the form

\textstyle{{\color[rgb]{1,0,0}\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\textstyle{{\color[rgb]{1,0,0}\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\textstyle{{\color[rgb]{1,0,0}\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\textstyle{{\color[rgb]{1,0,0}\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\textstyle{{\color[rgb]{1,0,0}\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\textstyle{{\color[rgb]{1,0,0}\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\textstyle{{\color[rgb]{1,0,0}\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\textstyle{{\color[rgb]{1,0,0}\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\textstyle{\bullet}

with the image of ϕ(1)×[1]italic-ϕ1delimited-[]1\phi(1)\times[1] indicated in red.

Therefore for each UΠ(f(1))op𝑈Πsuperscript𝑓1opU\in\Pi({f(1)})^{\rm op} the functor α(θ)1,[1](U,)𝛼subscript𝜃1delimited-[]1𝑈\alpha(\theta)_{1,[1]}(U,-) is the right Kan extension of a diagram in (FinΔ)opsuperscriptsubscriptFinΔop({\rm Fin}_{\Delta})^{\rm op} of the form

\textstyle{\bullet}\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\textstyle{\bullet}\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\textstyle{\bullet}\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\textstyle{\bullet}\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\bullet}\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\cdots}\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\bullet}\textstyle{\bullet}\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\textstyle{\bullet}\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\cdots}\textstyle{\bullet}\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\textstyle{\bullet}

As the equivalences in 𝒮pf(1),ϕ(1)𝒮subscriptp𝑓1italic-ϕ1\mathcal{S}{\rm p}_{f(1),\phi(1)} are exactly those cartesian functors Π(f(1))op×Σϕ(1),[1](FinΔ)opΠsuperscript𝑓1opsuperscriptΣitalic-ϕ1delimited-[]1superscriptsubscriptFinΔop\Pi({f(1)})^{\rm op}\times\Sigma^{\phi(1),[1]}\to({\rm Fin}_{\Delta})^{\rm op} which send all morphisms of the form (Id,Id,v)IdId𝑣({\rm Id},{\rm Id},v) to equivalences ([10] 6.2) the claimed result follows. ∎

Remark 4.7.

By Remark 2.7, the opposite of α𝛼\alpha gives a symmetric monoidal lax functor

χ=αop:𝒞oalg(𝒮pan2×((inΔ)op))op.:𝜒superscript𝛼op𝒞superscriptoalgcoproductsuperscript𝒮superscriptsubscriptpan2superscriptsubscriptindouble-struck-Δopop\chi=\alpha^{\rm op}:\mathcal{C}{\rm oalg}^{\amalg}\rightsquigarrow\left(\mathcal{S}{\rm pan}_{2}^{\times}\left({(\mathcal{F}{\rm in}_{\mathbb{\Delta}})^{\rm op}}\right)\right)^{\rm op}.

However, Corollary 3.12 says that (𝒮pan2×((inΔ)op))op𝒮pan2×((inΔ)op)similar-to-or-equalssuperscript𝒮superscriptsubscriptpan2superscriptsubscriptindouble-struck-Δopop𝒮superscriptsubscriptpan2superscriptsubscriptindouble-struck-Δop\left(\mathcal{S}{\rm pan}_{2}^{\times}\left({(\mathcal{F}{\rm in}_{\mathbb{\Delta}})^{\rm op}}\right)\right)^{\rm op}\simeq\mathcal{S}{\rm pan}_{2}^{\times}\left({(\mathcal{F}{\rm in}_{\mathbb{\Delta}})^{\rm op}}\right). Therefore, as a dual to Proposition 4.6, we obtain a lax coalgebra structure on Δ[1]Δdelimited-[]1\Delta\left[{1}\right].

4.2 The lax algebra structure inherited from Δ[1]Δdelimited-[]1\Delta\left[{1}\right]

Let 𝒳𝒞Δ𝒳subscript𝒞double-struck-Δ\mathcal{X}\in\mathcal{C}_{\mathbb{\Delta}} be a simplicial object in an \infty-category having finite limits. We can now show how 𝒳1subscript𝒳1\mathcal{X}_{1} inherits a lax algebra structure in 𝒮pan2×(𝒞)𝒮superscriptsubscriptpan2𝒞\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right) from the one endowed upon standard 111-simplex Δ[1]Δdelimited-[]1\Delta\left[{1}\right] in Proposition 4.6.

We make use of the following result due to Li-Bland.

Theorem 4.8 ([17] 4.1).

The construction which assigns to an \infty-category having finite limits its symmetric monoidal (,2)2(\infty,2)-category of bispans defines a functor

𝒮pan2×():𝒞atlex𝒞at(,2).:𝒮superscriptsubscriptpan2𝒞superscriptsubscriptatlex𝒞superscriptsubscriptat2tensor-product\mathcal{S}{\rm pan}_{2}^{\times}\left({-}\right):\mathcal{C}{\rm at}_{\infty}^{\rm lex}\to\mathcal{C}{\rm at}_{(\infty,2)}^{\otimes}.

where 𝒞atlex𝒞superscriptsubscriptatlex\mathcal{C}{\rm at}_{\infty}^{\rm lex} is the \infty-category of \infty-categories having finite limits and finite limit preserving functors between them.

Recall that any simplicial object 𝒳𝒞Δsubscript𝒳subscript𝒞double-struck-Δ\mathcal{X}_{\bullet}\in\mathcal{C}_{\mathbb{\Delta}} defines a finite limit preserving functor 𝒳:(inΔ)op𝒞:𝒳superscriptsubscriptindouble-struck-Δop𝒞\mathcal{X}:(\mathcal{F}{\rm in}_{\mathbb{\Delta}})^{\rm op}\to\mathcal{C} by right Kan extension,

(inΔ)opsuperscriptsubscriptindouble-struck-Δop\textstyle{(\mathcal{F}{\rm in}_{\mathbb{\Delta}})^{\rm op}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}𝒳𝒳\scriptstyle{\mathcal{X}}𝒞𝒞\textstyle{\mathcal{C}}Δopsuperscriptdouble-struck-Δop\textstyle{\mathbb{\Delta}^{\rm op}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}𝒳𝒳\scriptstyle{\mathcal{X}}

By Theorem 4.8 this induces a symmetric monoidal functor

𝒳:𝒮pan2×((inΔ)op)𝒮pan2×(𝒞).:𝒳𝒮superscriptsubscriptpan2superscriptsubscriptindouble-struck-Δop𝒮superscriptsubscriptpan2𝒞\mathcal{X}:\mathcal{S}{\rm pan}_{2}^{\times}\left({(\mathcal{F}{\rm in}_{\mathbb{\Delta}})^{\rm op}}\right)\to\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right).

Then as an immediate corollary of Proposition 4.6 one has the following.

Theorem 4.9.

Let 𝒞𝒞\mathcal{C} be an \infty-category with finite limits and let 𝒳𝒞Δsubscript𝒳subscript𝒞double-struck-Δ\mathcal{X}_{\bullet}\in\mathcal{C}_{\mathbb{\Delta}} be a simplicial object. Then the composite

α𝒳:𝒜lgα𝒮pan2×((inΔ)op)𝒳𝒮pan2×(𝒞),:subscript𝛼𝒳𝒜superscriptlgcoproduct𝛼𝒮superscriptsubscriptpan2superscriptsubscriptindouble-struck-Δop𝒳𝒮superscriptsubscriptpan2𝒞\alpha_{\mathcal{X}}:\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 12.7389pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\crcr}}}\ignorespaces{\hbox{\kern-12.7389pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathcal{A}{\rm lg}^{\amalg}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 19.49995pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\alpha}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 36.7389pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}{\hbox{\kern 36.7389pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathcal{S}{\rm pan}_{2}^{\times}\left({(\mathcal{F}{\rm in}_{\mathbb{\Delta}})^{\rm op}}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 114.57518pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.39166pt\hbox{$\scriptstyle{\mathcal{X}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 132.20018pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 132.20018pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right)}$}}}}}}}\ignorespaces}}}}\ignorespaces,

is a symmetric monoidal lax functor endowing 𝒳1subscript𝒳1\mathcal{X}_{1} with the structure of a lax algebra.

Remark 4.10.

Following Remark 4.7 we obtain a symmetric monoidal lax functor χ𝒳subscript𝜒𝒳\chi_{\mathcal{X}} endowing 𝒳1subscript𝒳1\mathcal{X}_{1} with the structure of a lax coalgebra.

4.3 Associativity and the 222-Segal condition

Having equipped the object of 111-simplices 𝒳1subscript𝒳1\mathcal{X}_{1} of a simplicial object 𝒳𝒞Δsubscript𝒳subscript𝒞double-struck-Δ\mathcal{X}_{\bullet}\in\mathcal{C}_{\mathbb{\Delta}} with a lax algebra structure in 𝒮pan2×(𝒞)𝒮superscriptsubscriptpan2𝒞\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right) in Theorem 4.9, we now demonstrate that the 222-Segal condition is exactly the right condition that enforces the associativity of this structure.

Definition 4.11.

[5] 2.3.2 A simplicial object 𝒳𝒞Δ𝒳subscript𝒞double-struck-Δ\mathcal{X}\in\mathcal{C}_{\mathbb{\Delta}} is a 222-Segal object if and only if for every n3𝑛3n\geq 3 and every 0i<jn0𝑖𝑗𝑛0\leq i<j\leq n, the image of the squares

Δ[{0,i}]Δdelimited-[]0𝑖\textstyle{\Delta\left[{\{0,i\}}\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Δ[{0,i,,n}]Δdelimited-[]0𝑖𝑛\textstyle{\Delta\left[{\{0,i,\ldots,n\}}\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}andand\textstyle{{\rm and}}Δ[{j,n}]Δdelimited-[]𝑗𝑛\textstyle{\Delta\left[{\{j,n\}}\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Δ[{0,,j}]Δdelimited-[]0𝑗\textstyle{\Delta\left[{\{0,\ldots,j\}}\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Δ[{0,,i}]Δdelimited-[]0𝑖\textstyle{\Delta\left[{\{0,\ldots,i\}}\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Δ[n]Δdelimited-[]𝑛\textstyle{\Delta\left[{n}\right]}Δ[{j,,n}]Δdelimited-[]𝑗𝑛\textstyle{\Delta\left[{\{j,\ldots,n\}}\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Δ[n]Δdelimited-[]𝑛\textstyle{\Delta\left[{n}\right]} (4.4)

under 𝒳𝒳\mathcal{X} are pullbacks in 𝒞𝒞\mathcal{C} and for each n2𝑛2n\geq 2 and 0i<n0𝑖𝑛0\leq i<n, the image of the square

Δ[{i,i+1}]Δdelimited-[]𝑖𝑖1\textstyle{\Delta\left[{\{i,i+1\}}\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Δ[n]Δdelimited-[]𝑛\textstyle{\Delta\left[{n}\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Δ[{i}]Δdelimited-[]𝑖\textstyle{\Delta\left[{\{i\}}\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Δ[n1]Δdelimited-[]𝑛1\textstyle{\Delta\left[{n-1}\right]} (4.5)

under 𝒳𝒳\mathcal{X} is a pullback in 𝒞𝒞\mathcal{C}. In particular, a 222-Segal space is a 222-Segal object in 𝒮𝒮\mathcal{S}, the \infty-category of spaces.

Remark 4.12.

What we call a 222-Segal object is called a unital 222-Segal object in [5] and a decomposition space in [7].

Remark 4.13.

For a simplicial object 𝒳𝒳\mathcal{X} to be 222-Segal it suffices for the images under 𝒳𝒳\mathcal{X} of the squares in Eq. 4.4 to be equivalences for i=0𝑖0i=0 or j=n𝑗𝑛j=n ([5] 2.3.2).

Theorem 4.14.

Let 𝒞𝒞\mathcal{C} be an \infty-category with finite limits. Then a simplicial object 𝒳𝒞Δ𝒳subscript𝒞double-struck-Δ\mathcal{X}\in\mathcal{C}_{\mathbb{\Delta}} is a 222-Segal object if and only if the symmetric monoidal lax functor α𝒳subscript𝛼𝒳\alpha_{\mathcal{X}} of Theorem 4.9 endows 𝒳1subscript𝒳1\mathcal{X}_{1} with the structure of a algebra.

Proof.

We must show that 𝒳𝒳\mathcal{X} is a 222-Segal object if and only if α𝒳(θ)1,[1]subscript𝛼𝒳subscript𝜃1delimited-[]1\alpha_{\mathcal{X}}(\theta)_{1,[1]} is an equivalence in 𝒮pf(1),ϕ(1)(𝒞)𝒮subscriptp𝑓1italic-ϕ1𝒞\mathcal{S}{\rm p}_{f(1),\phi(1)}(\mathcal{C}) for every ((f,ϕ),θ)Un(𝒜lg)1𝑓italic-ϕ𝜃Unsubscript𝒜superscriptlgcoproduct1\left((f,\phi),\theta\right)\in{\rm Un}(\mathcal{A}{\rm lg}^{\amalg})_{1}.

Every active morphism ψ:[n]|[m]\psi:[n]\rightarrow\Mapsfromchar[m] can be decomposed as

i=1nψi:i=1n[1]i=1n[mi],:superscriptsubscript𝑖1𝑛subscript𝜓𝑖superscriptsubscript𝑖1𝑛delimited-[]1superscriptsubscript𝑖1𝑛delimited-[]subscript𝑚𝑖\bigvee_{i=1}^{n}\psi_{i}:\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 17.40352pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-17.40352pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\displaystyle\bigvee_{i=1}^{n}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 34.20345pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\lower 0.0pt\hbox{\lx@xy@stopper}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 34.20345pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\displaystyle\bigvee_{i=1}^{n}[m_{i}]}$}}}}}}}\ignorespaces}}}}\ignorespaces,

where ψisubscript𝜓𝑖\psi_{i} is the unique active morphism [1]|[mi][1]\rightarrow\Mapsfromchar[m_{i}] and [m]=i[mi]delimited-[]𝑚subscript𝑖delimited-[]subscript𝑚𝑖[m]=\vee_{i}[m_{i}]. Every inert morphism is of the form [n][a][n][b]delimited-[]𝑛delimited-[]𝑎delimited-[]𝑛delimited-[]𝑏[n]\rightarrowtail[a]\vee[n]\vee[b]. Since morphisms in ΔΔ\Delta can be uniquely factored as an active morphism followed by an inert morphism, one can decompose the morphism ϕ:[m][n]N1(Δop):italic-ϕdelimited-[]𝑚delimited-[]𝑛subscript𝑁1superscriptΔop\phi:[m]\leftarrow[n]\in N_{1}(\Delta^{\rm op}) as

ϕ=i=1nϕi:[aϕ](i=1n[mi])[bϕ]i=1n[1].:italic-ϕsuperscriptsubscript𝑖1𝑛subscriptitalic-ϕ𝑖delimited-[]subscript𝑎italic-ϕsuperscriptsubscript𝑖1𝑛delimited-[]subscript𝑚𝑖delimited-[]subscript𝑏italic-ϕsuperscriptsubscript𝑖1𝑛delimited-[]1\phi=\bigvee_{i=1}^{n}\phi_{i}:[a_{\phi}]\vee\left(\bigvee_{i=1}^{n}[m_{i}]\right)[b_{\phi}]\leftarrow\bigvee_{i=1}^{n}[1].

One can therefore write the poset Mϕsubscript𝑀italic-ϕM_{\phi} schematically as

[Uncaptioned image]

Denoting by θi:Π(f(0))×[1]Alg:subscript𝜃𝑖Π𝑓0delimited-[]1Alg\theta_{i}:\Pi({f(0)})\times[1]\to{\rm Alg} the restriction of θ𝜃\theta to the i𝑖i-th summand of ϕ(1)italic-ϕ1\phi(1), it follows that for each UΠ(f(1))op𝑈Πsuperscript𝑓1opU\in\Pi({f(1)})^{\rm op}, the functor α𝒳(θ)ϕ(1),[1](U,)subscript𝛼𝒳subscript𝜃italic-ϕ1delimited-[]1𝑈\alpha_{\mathcal{X}}(\theta)_{\phi(1),[1]}(U,-) is the right Kan extension of a diagram in 𝒞𝒞\mathcal{C} of the form

[Uncaptioned image]

Therefore α𝒳(θ)1,[1]subscript𝛼𝒳subscript𝜃1delimited-[]1\alpha_{\mathcal{X}}(\theta)_{1,[1]} is an equivalence if and only if α𝒳(θi)1,[1]subscript𝛼𝒳subscriptsubscript𝜃𝑖1delimited-[]1\alpha_{\mathcal{X}}(\theta_{i})_{1,[1]} is an equivalence for each i𝑖i.

We conclude that α𝒳:𝒜lg𝒮pan2×(𝒞):subscript𝛼𝒳𝒜superscriptlgcoproduct𝒮superscriptsubscriptpan2𝒞\alpha_{\mathcal{X}}:\mathcal{A}{\rm lg}^{\amalg}\rightsquigarrow\mathcal{S}{\rm pan}_{2}^{\times}\left({\mathcal{C}}\right) is a symmetric monoidal functor if and only if α𝒳(θ)1,[1]subscript𝛼𝒳subscript𝜃1delimited-[]1\alpha_{\mathcal{X}}(\theta)_{1,[1]} is an equivalence in 𝒮pf(1),ϕ(1)(𝒞)𝒮subscriptp𝑓1italic-ϕ1𝒞\mathcal{S}{\rm p}_{f(1),\phi(1)}(\mathcal{C}) for every ((f,ϕ),θ)Un(𝒜lg)1𝑓italic-ϕ𝜃Unsubscript𝒜superscriptlgcoproduct1\left((f,\phi),\theta\right)\in{\rm Un}(\mathcal{A}{\rm lg}^{\amalg})_{1} where ϕ:[n]|[1]\phi:[n]\Mapstochar\leftarrow[1] is the unique active morphism. By Lemmas 4.15 and 4.16, proven below, this latter condition is equivalent to 𝒳𝒳\mathcal{X} satisfying the 222-Segal condition. ∎

To conclude our proof of Theorem 4.14 we must prove two lemmas.

Lemma 4.15.

Let 𝒳𝒞Δ𝒳subscript𝒞double-struck-Δ\mathcal{X}\in\mathcal{C}_{\mathbb{\Delta}} be a simplicial object in an \infty-category having finite limits. If, for every ((f,ϕ),θ)Un(𝒜lg)1𝑓italic-ϕ𝜃Unsubscript𝒜superscriptlgcoproduct1\left((f,\phi),\theta\right)\in{\rm Un}(\mathcal{A}{\rm lg}^{\amalg})_{1} with ϕ:[n]|[1]\phi:[n]\Mapstochar\leftarrow[1] being the unique active morphism α𝒳(θ)1,[1]subscript𝛼𝒳subscript𝜃1delimited-[]1\alpha_{\mathcal{X}}(\theta)_{1,[1]} is an equivalence in 𝒮pf(1),ϕ(1)(𝒞)𝒮subscriptp𝑓1italic-ϕ1𝒞\mathcal{S}{\rm p}_{f(1),\phi(1)}(\mathcal{C}), then 𝒳𝒳\mathcal{X} satisfies the 222-Segal condition.

Proof.

Consider, for each n3𝑛3n\geq 3 and i,j=1,,n1formulae-sequence𝑖𝑗1𝑛1i,j=1,\ldots,n-1, the following pairs of composable morphisms in AlgAlg{\rm Alg}:

{1,,n}1𝑛\textstyle{\{1,\ldots,n\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}p1isubscriptsuperscript𝑝𝑖1\scriptstyle{p^{i}_{1}}{i,,n}𝑖𝑛\textstyle{\{i,\ldots,n\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}p2isubscriptsuperscript𝑝𝑖2\scriptstyle{p^{i}_{2}}1¯¯1\textstyle{\underline{1}}{1,,n}1𝑛\textstyle{\{1,\ldots,n\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}p1jsubscriptsuperscript𝑝𝑗1\scriptstyle{p^{j}_{1}}{1,,j+1}1𝑗1\textstyle{\{1,\ldots,j+1\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}p2jsubscriptsuperscript𝑝𝑗2\scriptstyle{p^{j}_{2}}1¯.¯1\textstyle{\underline{1}\ .} (4.6)

The morphism p1isubscriptsuperscript𝑝𝑖1p^{i}_{1} maps a𝑎a to i𝑖i when 1ai1𝑎𝑖1\leq a\leq i and a𝑎a otherwise, while p1jsubscriptsuperscript𝑝𝑗1p^{j}_{1} maps a𝑎a to j+1𝑗1j+1 when j+1an𝑗1𝑎𝑛j+1\leq a\leq n and a𝑎a otherwise. The linear orders on the fibres are induced by the ordering on {1,,n}1𝑛\{1,\ldots,n\}. Observe that

α(p2i)α({i,,n})α(p1i)andα(p2j)α({1,,j+1})α(p1j)𝛼subscriptsuperscript𝑝𝑖2subscriptcoproduct𝛼𝑖𝑛𝛼subscriptsuperscript𝑝𝑖1and𝛼subscriptsuperscript𝑝𝑗2subscriptcoproduct𝛼1𝑗1𝛼subscriptsuperscript𝑝𝑗1\alpha(p^{i}_{2})\coprod_{\alpha(\{i,\ldots,n\})}\alpha(p^{i}_{1})\ \ {\rm and}\ \ \alpha(p^{j}_{2})\coprod_{\alpha(\{1,\ldots,j+1\})}\alpha(p^{j}_{1})

are, respectively, the pushouts of the left and right square of Eq. 4.4. Furthermore, α(p2ip1i)=Δ[n]=α(p2jp1j)𝛼subscriptsuperscript𝑝𝑖2subscriptsuperscript𝑝𝑖1Δdelimited-[]𝑛𝛼subscriptsuperscript𝑝𝑗2subscriptsuperscript𝑝𝑗1\alpha(p^{i}_{2}p^{i}_{1})=\Delta\left[{n}\right]=\alpha(p^{j}_{2}p^{j}_{1}), and the corresponding components of the lax structure on α𝛼\alpha agree with the ones arising from the squares in Eq. 4.4.

Next, consider for each n2𝑛2n\geq 2 and 0i<n0𝑖𝑛0\leq i<n the pair of composable morphisms in AlgAlg{\rm Alg}:

{1,,n1}1𝑛1\textstyle{\{1,\ldots,n-1\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}e1isubscriptsuperscript𝑒𝑖1\scriptstyle{e^{i}_{1}}{1,,n}1𝑛\textstyle{\{1,\ldots,n\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}e2isubscriptsuperscript𝑒𝑖2\scriptstyle{e^{i}_{2}}1¯,¯1\textstyle{\underline{1}\ ,} (4.7)

where e1isubscriptsuperscript𝑒𝑖1e^{i}_{1} is the evident map which skips the element i+1𝑖1i+1 in {1,,n}1𝑛\{1,\ldots,n\}. Observe that

α(e2i)α({1,,n})α(e1i)𝛼subscriptsuperscript𝑒𝑖2subscriptcoproduct𝛼1𝑛𝛼subscriptsuperscript𝑒𝑖1\alpha(e^{i}_{2})\coprod_{\alpha(\{1,\ldots,n\})}\alpha(e^{i}_{1})

is the pushout of Eq. 4.5. We also have that α(e2ie1i)=Δ[n1]𝛼subscriptsuperscript𝑒𝑖2subscriptsuperscript𝑒𝑖1Δdelimited-[]𝑛1\alpha(e^{i}_{2}e^{i}_{1})=\Delta\left[{n-1}\right], and the corresponding components of the lax structure on α𝛼\alpha agree with the ones arising from the square in Eq. 4.5.

Finally, consider ((f,ϕ),θ)Un(𝒜lg)1𝑓italic-ϕ𝜃Unsubscript𝒜superscriptlgcoproduct1\left((f,\phi),\theta\right)\in{\rm Un}(\mathcal{A}{\rm lg}^{\amalg})_{1}, where f𝑓f is constant on 1¯subscript¯1\underline{1}_{\ast} and ϕ:[2]|[1]\phi:[2]\Mapstochar\leftarrow[1] is the unique active map. Then the functor θ𝜃\theta is a pair of composable morphisms

X0subscript𝑋0\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}p1subscript𝑝1\scriptstyle{p_{1}}X1subscript𝑋1\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}p2subscript𝑝2\scriptstyle{p_{2}}X2subscript𝑋2\textstyle{X_{2}}

and α𝒳(θ)1,[1]subscript𝛼𝒳subscript𝜃1delimited-[]1\alpha_{\mathcal{X}}(\theta)_{1,[1]} is the diagram

α𝒳(X1)subscript𝛼𝒳subscript𝑋1\textstyle{\alpha_{\mathcal{X}}(X_{1})}α𝒳(p2p1)subscript𝛼𝒳subscript𝑝2subscript𝑝1\textstyle{\alpha_{\mathcal{X}}(p_{2}p_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α𝒳(X2)subscript𝛼𝒳subscript𝑋2\textstyle{\alpha_{\mathcal{X}}(X_{2})}α𝒳(X1)subscript𝛼𝒳subscript𝑋1\textstyle{\alpha_{\mathcal{X}}(X_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α𝒳(p2p1)subscript𝛼𝒳subscript𝑝2subscript𝑝1\textstyle{\alpha_{\mathcal{X}}(p_{2}p_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}δ𝛿\scriptstyle{\delta}α𝒳(X2)subscript𝛼𝒳subscript𝑋2\textstyle{\alpha_{\mathcal{X}}(X_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α𝒳(X1)subscript𝛼𝒳subscript𝑋1\textstyle{\alpha_{\mathcal{X}}(X_{1})}α𝒳(p2)×α𝒳(X1)α𝒳(p1)subscriptsubscript𝛼𝒳subscript𝑋1subscript𝛼𝒳subscript𝑝2subscript𝛼𝒳subscript𝑝1\textstyle{\alpha_{\mathcal{X}}(p_{2})\times_{\alpha_{\mathcal{X}}(X_{1})}\alpha_{\mathcal{X}}(p_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}α𝒳(X2)subscript𝛼𝒳subscript𝑋2\textstyle{\alpha_{\mathcal{X}}(X_{2})}

The functor α𝒳(θ)1,[1]subscript𝛼𝒳subscript𝜃1delimited-[]1\alpha_{\mathcal{X}}(\theta)_{1,[1]} is an equivalence in 𝒮p1¯,[1]𝒮subscriptp¯1delimited-[]1\mathcal{S}{\rm p}_{\underline{1},[1]} precisely when δ𝛿\delta is an equivalence.

Taking θ𝜃\theta to be the pairs of morphisms defined in Eqs. 4.6 and 4.7 it follows that 𝒳𝒳\mathcal{X} is a 222-Segal object. ∎

For the second of the two lemmas which complete Theorem 4.14 we must make use of an equivalent formulation of the 222-Segal condition due to Gálvez-Carrillo–Kock–Tonks ([7] 3). They show that 𝒳𝒞Δ𝒳subscript𝒞double-struck-Δ\mathcal{X}\in\mathcal{C}_{\mathbb{\Delta}} is 222-Segal if and only if the image under 𝒳𝒳\mathcal{X} of every pushout square in ΔΔ\Delta of the form

[n]delimited-[]𝑛\textstyle{[n]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}[m]delimited-[]𝑚\textstyle{[m]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}[k]delimited-[]𝑘\textstyle{[k]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}[l]delimited-[]𝑙\textstyle{[l]}

is a pullback in 𝒞𝒞\mathcal{C}.

Lemma 4.16.

Let 𝒳𝒞Δ𝒳subscript𝒞double-struck-Δ\mathcal{X}\in\mathcal{C}_{\mathbb{\Delta}} be a 222-Segal object. Then α𝒳(θ)1,[1]subscript𝛼𝒳subscript𝜃1delimited-[]1\alpha_{\mathcal{X}}(\theta)_{1,[1]} is an equivalence in 𝒮pf(1),ϕ(1)(𝒞)𝒮subscriptp𝑓1italic-ϕ1𝒞\mathcal{S}{\rm p}_{f(1),\phi(1)}(\mathcal{C}) for every ((f,ϕ),θ)Un(𝒜lg)1𝑓italic-ϕ𝜃Unsubscript𝒜superscriptlgcoproduct1\left((f,\phi),\theta\right)\in{\rm Un}(\mathcal{A}{\rm lg}^{\amalg})_{1} with ϕ:[n]|[1]\phi:[n]\Mapstochar\leftarrow[1] being the unique active morphism.

Proof.

It suffices to show that for every sequence of composable morphisms in AlgAlg{\rm Alg}

X0p1X1p2pnXn,subscript𝑋0subscript𝑝1subscript𝑋1subscript𝑝2subscript𝑝𝑛subscript𝑋𝑛\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 8.93471pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\crcr}}}\ignorespaces{\hbox{\kern-8.93471pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 15.17377pt\raise 5.1875pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8264pt\hbox{$\scriptstyle{p_{1}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 32.93471pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 32.93471pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 57.04318pt\raise 5.1875pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8264pt\hbox{$\scriptstyle{p_{2}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 74.80412pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 74.80412pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 95.5754pt\raise 5.1875pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8264pt\hbox{$\scriptstyle{p_{n}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 112.30412pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 112.30412pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{X_{n}}$}}}}}}}\ignorespaces}}}}\ignorespaces,

the image of

α(pn)α(Xn1)α(X1)α(p1)α(pnp1)𝛼subscript𝑝𝑛subscriptcoproduct𝛼subscript𝑋𝑛1subscriptcoproduct𝛼subscript𝑋1𝛼subscript𝑝1𝛼subscript𝑝𝑛subscript𝑝1\alpha(p_{n})\coprod_{\alpha(X_{n-1})}\cdots\coprod_{\alpha(X_{1})}\alpha(p_{1})\to\alpha(p_{n}\cdots p_{1})

under 𝒳𝒳\mathcal{X} is an equivalence. Since every morphism in AlgAlg{\rm Alg} is a disjoint union of morphisms having target the singleton set, and the functors α𝛼\alpha and 𝒳𝒳\mathcal{X} are symmetric monoidal, it suffices to consider the case when Xn=1¯subscript𝑋𝑛¯1X_{n}=\underline{1}. The statement for general n𝑛n follows by iterating the special case of n=2𝑛2n=2.

First, in the trivial case of X1=0¯subscript𝑋1¯0X_{1}=\underline{0}, then also X0=0¯subscript𝑋0¯0X_{0}=\underline{0} and

α(p2)α(X1)α(p1)=Δ[0]=α(p2p1).𝛼subscript𝑝2subscriptcoproduct𝛼subscript𝑋1𝛼subscript𝑝1Δdelimited-[]0𝛼subscript𝑝2subscript𝑝1\alpha(p_{2})\coprod_{\alpha(X_{1})}\alpha(p_{1})=\Delta\left[{0}\right]=\alpha(p_{2}p_{1}).

For X10¯subscript𝑋1¯0X_{1}\neq\underline{0}, write X1={1,,k}subscript𝑋11𝑘X_{1}=\{1,\ldots,k\} and denote by

κ=𝔊1(p2)andκi=𝔊1(p1|p11(i)),iX1,formulae-sequence𝜅superscript𝔊1subscript𝑝2andsubscript𝜅𝑖superscript𝔊1evaluated-atsubscript𝑝1superscriptsubscript𝑝11𝑖𝑖subscript𝑋1\kappa=\mathfrak{G}^{-1}(p_{2})\ {\rm and}\ \kappa_{i}=\mathfrak{G}^{-1}\left(p_{1}|_{p_{1}^{-1}(i)}\right),\ i\in X_{1},

where 𝔊1(n)=[n]superscript𝔊1delimited-⟨⟩𝑛delimited-[]𝑛\mathfrak{G}^{-1}(\langle n\rangle)=[n] is the inverse on objects of the functor in Eq. 1.3. Recall that Δ[κ]Δdelimited-[]𝜅\Delta\left[{\kappa}\right] can be thought of as a standard simplex having the edges along its spine labelled by the elements of X1subscript𝑋1X_{1} according to the linear ordering defined by p2subscript𝑝2p_{2}. The pushout α(p2)α(X1)α(p1)𝛼subscript𝑝2subscriptcoproduct𝛼subscript𝑋1𝛼subscript𝑝1\alpha(p_{2})\coprod_{\alpha(X_{1})}\alpha(p_{1}) is the simplicial set obtained by gluing each simplex Δ[κi]Δdelimited-[]subscript𝜅𝑖\Delta\left[{\kappa_{i}}\right] by its long edge to the corresponding edge on the spine of Δ[κ]Δdelimited-[]𝜅\Delta\left[{\kappa}\right]. Therefore α(p2)α(X1)α(p1)𝛼subscript𝑝2subscriptcoproduct𝛼subscript𝑋1𝛼subscript𝑝1\alpha(p_{2})\coprod_{\alpha(X_{1})}\alpha(p_{1}) is the iterated pushout

Δ[κ]i=1kΔ[κi]=(((Δ[κ]Δ[(0,1)]Δ[κ1])Δ[(1,2)]Δ[κ2]))Δ[(k1,k)]Δ[κk].Δdelimited-[]𝜅superscriptsubscriptcoproduct𝑖1𝑘Δdelimited-[]subscript𝜅𝑖Δdelimited-[]𝜅subscriptcoproductΔdelimited-[]01Δdelimited-[]subscript𝜅1subscriptcoproductΔdelimited-[]12Δdelimited-[]subscript𝜅2subscriptcoproductΔdelimited-[]𝑘1𝑘Δdelimited-[]subscript𝜅𝑘\Delta\left[{\kappa}\right]\coprod_{i=1}^{k}\Delta\left[{\kappa_{i}}\right]=\left(\cdots\left(\left(\Delta\left[{\kappa}\right]\coprod_{\Delta\left[{(0,1)}\right]}\Delta\left[{\kappa_{1}}\right]\right)\coprod_{\Delta\left[{(1,2)}\right]}\Delta\left[{\kappa_{2}}\right]\right)\cdots\right)\coprod_{\Delta\left[{(k-1,k)}\right]}\Delta\left[{\kappa_{k}}\right]\ .

Next, set κisubscript𝜅absent𝑖\kappa_{\leq i} to be the inductively defined pushouts in ΔΔ\Delta

[1]delimited-[]1\textstyle{[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}g0subscript𝑔0\scriptstyle{g_{0}}κ1subscript𝜅1\textstyle{\kappa_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}andand\textstyle{{\rm and}}[1]delimited-[]1\textstyle{[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}gisubscript𝑔𝑖\scriptstyle{g_{i}}κi+1subscript𝜅𝑖1\textstyle{\kappa_{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}κ𝜅\textstyle{\kappa\ignorespaces\ignorespaces\ignorespaces\ignorespaces}κ1subscript𝜅absent1\textstyle{\kappa_{\leq 1}}κisubscript𝜅absent𝑖\textstyle{\kappa_{\leq i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}κi+1subscript𝜅absent𝑖1\textstyle{\kappa_{\leq i+1}} (4.8)

where g0subscript𝑔0g_{0} has image the smallest two elements of κ𝜅\kappa, and gisubscript𝑔𝑖g_{i} has image the kisubscript𝑘𝑖k_{i}’th and ki+1subscript𝑘𝑖1k_{i}+1’th elements of κisubscript𝜅absent𝑖\kappa_{\leq i} where ki=j=1i|κj|1subscript𝑘𝑖superscriptsubscript𝑗1𝑖subscript𝜅𝑗1k_{i}=\sum_{j=1}^{i}|\kappa_{j}|-1. Then since κk=i=1kκisubscript𝜅absent𝑘superscriptsubscript𝑖1𝑘subscript𝜅𝑖\kappa_{\leq k}=\vee_{i=1}^{k}\kappa_{i} one has that Δ[κk]=α(p2p1)Δdelimited-[]subscript𝜅absent𝑘𝛼subscript𝑝2subscript𝑝1\Delta\left[{\kappa_{\leq k}}\right]=\alpha(p_{2}p_{1}). Furthermore, the morphism α(p2)α(X1)α(p1)α(p2p1)𝛼subscript𝑝2subscriptcoproduct𝛼subscript𝑋1𝛼subscript𝑝1𝛼subscript𝑝2subscript𝑝1\alpha(p_{2})\coprod_{\alpha(X_{1})}\alpha(p_{1})\to\alpha(p_{2}p_{1}) factors as the composite

Δ[κ]i=1kΔ[κi]Δdelimited-[]𝜅superscriptsubscriptcoproduct𝑖1𝑘Δdelimited-[]subscript𝜅𝑖\textstyle{\Delta\left[{\kappa}\right]\displaystyle\coprod_{i=1}^{k}\Delta\left[{\kappa_{i}}\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Δ[κ1]i=2kΔ[κi]Δdelimited-[]subscript𝜅absent1superscriptsubscriptcoproduct𝑖2𝑘Δdelimited-[]subscript𝜅𝑖\textstyle{\Delta\left[{\kappa_{\leq 1}}\right]\displaystyle\coprod_{i=2}^{k}\Delta\left[{\kappa_{i}}\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Δ[κk]Δdelimited-[]subscript𝜅absent𝑘\textstyle{\Delta\left[{\kappa_{\leq k}}\right]\,} (4.9)

where each morphism arises from the pushout squares in Eq. 4.8.

It follows from the Gálvez-Carrillo–Kock–Tonks form of the 222-Segal condition that each morphism the sequence in Eq. 4.9 is sent to an equivalence under 𝒳𝒳\mathcal{X}, proving the claimed result. ∎

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